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All Textbook Solutions for Differential Equations

In Exercises 1 and 2, find the equilibrium solutions of the differential equation specified. 1. dydt=y+31y In Exercises 1 and 2, find the equilibrium solutions of the differential equation specified. 2. dydt=(t21)(y22)y24 Consider the population model dPdt=0.4P(1P230) where P(t) is the population at time t. (a) For what values of P is the population in equilibrium? (b) For what values of P is the population increasing? (c) For what values of P is the population decreasing?Consider the population model dPdt=0.3(1P200)(P501)P where P(t) is the population at time t. (a) For what values of P is the population in equilibrium? (b) For what values of P is the population increasing? (c) For what values of P is the population decreasing?Consider the differential equation dydt=y3y212y (a) For what values of y is y(t) in equilibrium? (b) For what values of y is y(t) increasing? (c) For what values of y is y(t) decreasing?In Exercises 6—10, we consider the phenomenon of radioactive decay which, from experimentation, we know behaves according to the law: The rate at which a quantity of a radioactive isotope decays is proportional to the amount of the isotope present. The proportionality constant depends only on which radioactive isotope is used. 6. Model radioactive decay using the notation t = time (independent variable), r(t) = amount of particular radioactive isotope present at time t (dependent variable), = decay rate (parameter). Note that the minus sign is used so that > 0. (a) Using this notation, write a model for the decay of a particular radioactive isotope. (b) If the amount of the isotope present at t = 0 is r0, state the corresponding initial-value problem for the model in part (a).In Exercises 6—10, we consider the phenomenon of radioactive decay which, from experimentation, we know behaves according to the law: The rate at which a quantity of a radioactive isotope decays is proportional to the amount of the isotope present. The proportionality constant depends only on which radioactive isotope is used. 7. The half-life of a radioactive isotope is the amount of time it takes for a quantity of radioactive material to decay to one-half of its original amount. (a) The half-life of Carbon 14 (C-14) is 5230 years. Determine the decay-rate pa-rameter for C-14. (b) The half-life of Iodine 131 (I-131) is 8 days. Determine the decay-rate param-eter for I-131. (c) What are the units of the decay-rate parameters in parts (a) and (b)? (d) To determine the half-life of an isotope, we could start with atoms of the isotope and measure the amount of time it takes 500 of them to decay, or we could start with 10,000 atoms of the isotope and measure the amount of time it takes 5000 of them to decay. Will we get the same answer? Why?In Exercises 6—10, we consider the phenomenon of radioactive decay which, from experimentation, we know behaves according to the law: The rate at which a quantity of a radioactive isotope decays is proportional to the amount of the isotope present. The proportionality constant depends only on which radioactive isotope is used. 8. Carbon dating is a method of determining the time elapsed since the death of organicmaterial. The assumptions implicit in carbon dating are that • Carbon 14 (C-14) makes up a constant proportion of the carbon that living matter ingests on a regular basis, and • once the matter dies, the C-14 present decays, but no new carbon is added tothe matter. Hence, by measuring the amount of C-14 still in the organic matter and comparing it to the amount of C-14 typically found in living matter, a "time since death" can be approximated. Using the decay-rate parameter you computed in Exercise 7, determine the time since death if (a) 88% of the original C-14 is still in the material. (b) 12% of the original C-14 is still in the material. (c) 2% of the original C-14 is still in the material. (d) 98% of the original C-14 is still in the material. Remark: There has been speculation that the amount of C-14 available to living creatures has not been exactly constant over long periods (thousands of years). This makes accurate dates much trickier to determine.In Exercises 6—10, we consider the phenomenon of radioactive decay which, from experimentation, we know behaves according to the law: The rate at which a quantity of a radioactive isotope decays is proportional to the amount of the isotope present. The proportionality constant depends only on which radioactive isotope is used. 9. Engineers and scientists often measure the rate of decay of an exponentially decaying quantity using its time constant. The time constant is the amount of time that an exponentially decaying quantity takes to decay by a factor of l/e. Because l/e is approximately 0.368, is the amount of time that the quantity takes to decay to approximately 36.8% of its original amount. (a) How are the time constant and the decay rate related? (b) Express the time constant in terms of the half-life. (c) What are the time constants for Carbon 14 and Iodine 131? (d) Given an exponentially decaying quantity r(t) with initial value r0= r(0), show that its time constant is the time at which the tangent line to the graph of r(t)/r0at (0, l) crosses the t-axis. [Hint: Start by sketching the graph of r(t)/r0and the line tangent to the graph at (0, 1).] (e) It is often said that an exponentially decaying quantity reaches its steady state in five time constants, that is, at t =5 . Explain why this statement is not literally true but is correct for all practical purposes.In Exercises 6—10, we consider the phenomenon of radioactive decay which, from experimentation, we know behaves according to the law: The rate at which a quantity of a radioactive isotope decays is proportional to the amount of the isotope present. The proportionality constant depends only on which radioactive isotope is used. 10. The radioactive isotope I-131 is used in the treatment of hyperthyroidism. When administered to a patient, I-131 accumulates in the thyroid gland, where it decays and kills part of that gland. (a) Suppose that it takes 72 hours to ship I-131 from the producer to the hospital. What percentage of the original amount shipped actually arrives at the hospital? (See Exercise 7.) (b) If the I- 131 is stored at the hospital for an additional 48 hours before it is used, how much of the original amount shipped from the producer is left when it is used? (c) How long will it take for the I-131 to decay completely so that the remnants can be thrown away without special precautions?MacQuarie Island is a small island about half-way between Antarctica and New Zealand. Between 2000 and 2006, the population of rabbits on the island rose from 4,000 to 130,000. Model the growth in the rabbit population R(t) at time t using an exponential growth model dRdt=kR where t = 0 corresponds to the year 2000. What is an appropriate value for the growth-rate parameter k, and what does this model predict for the population in the year 2010. (For more information on why the population of rabbits exploded, see Review Exercise 22 in Chapter 2.)The velocity u of a freefalling skydiver is well modeled by the differential equation mdvdt=mgkv2 where m is the mass of the skydiver, g is the gravitational constant, and k is the drag coefficient determined by the position of the diver during the dive. (Note that the constants m, g, and k are positive.) (a) Perform a qualitative analysis of this model. (b) Calculate the terminal velocity of the skydiver. Express your answer in termsof m, g, and k.Exercises 13—15 consider an elementary model of the learning process: Although human learning is an extremely complicated process, it is possible to build models of certain simple types of memorization. For example, consider a person presented with a list to be studied. The subject is given periodic quizzes to determine exactly how much of the list has been memorized. (The lists are usually things like nonsense syllables, randomly generated three-digit numbers, or entries from tables of integrals.) If we let L(t) be the fraction of the list learned at time t, where L = 0 corresponds to knowing nothing and L = 1 corresponds to knowing the entire list, then we can form a simple model of this type of learning based on the assumption: • The rate dL/dt is proportional to the fraction of the list left to be learned. Since L = 1 corresponds to knowing the entire list, the model is dLdt=k(1L) where k is the constant of proportionality. 13. For what value of L, 0L1 , does learning occur most rapidly?Exercises 13—15 consider an elementary model of the learning process: Although human learning is an extremely complicated process, it is possible to build models of certain simple types of memorization. For example, consider a person presented with a list to be studied. The subject is given periodic quizzes to determine exactly how much of the list has been memorized. (The lists are usually things like nonsense syllables, randomly generated three-digit numbers, or entries from tables of integrals.) If we let L(t) be the fraction of the list learned at time t, where L = 0 corresponds to knowing nothing and L = 1 corresponds to knowing the entire list, then we can form a simple model of this type of learning based on the assumption: • The rate dL/dt is proportional to the fraction of the list left to be learned. Since L = 1 corresponds to knowing the entire list, the model is dLdt=k(1L) where k is the constant of proportionality. 14. Suppose two students memorize lists according to the model dLdt=2(1L) (a) If one of the students knows one-half of the list at time t = 0 and the other knows none of the list, which student is learning more rapidly at this instant? (b) Will the student who starts out knowing none of the list ever catch up to the student who starts out knowing one-half of the list?Exercises 13—15 consider an elementary model of the learning process: Although human learning is an extremely complicated process, it is possible to build models of certain simple types of memorization. For example, consider a person presented with a list to be studied. The subject is given periodic quizzes to determine exactly how much of the list has been memorized. (The lists are usually things like nonsense syllables, randomly generated three-digit numbers, or entries from tables of integrals.) If we let L(t) be the fraction of the list learned at time t, where L = 0 corresponds to knowing nothing and L = 1 corresponds to knowing the entire list, then we can form a simple model of this type of learning based on the assumption: • The rate dL/dt is proportional to the fraction of the list left to be learned. Since L = 1 corresponds to knowing the entire list, the model is dLdt=k(1L) where k is the constant of proportionality. 15. Consider the following two differential equations that model two students' rates of memorizing a poem. Aly's rate is proportional to the amount to be learned with proportionality constant k = 2. Beth's rate is proportional to the square of the amount to be learned with proportionality constant 3. The corresponding differential equations dLAdt=2(1LA) and dLBdt=3(1LB)2 where LA(t) and LB(t) are the fractions of the poem learned at time t by Aly and Beth, respectively. (a) Which student has a faster rate of learning at t = 0 if they both stan memorizing together having never seen the poem before? (b) Which student has a faster rate of learning at t = 0 if they both stan memorizing together having already learned one-half of the poem? (c) Which student has a faster rate of learning at t = 0 if they both stan memorizing together having already learned one-third of the poem?The expenditure on education in the U.S. is given in the following table. (Amounts are expressed in millions of 2001 constant dollars.) (a) Let s(t)=s0ekt be an exponential function. Show that the graph of ln s(t) a function of t is a line. What is its slope and vertical intercept? (b) Is spending on education in the U.S. rising exponentially fast? If so, what is the growth-rate coefficient? [Hint: Use your solution to part (a).]Suppose a species of fish in a particular lake has a population that is modeled by the logistic population model with growth rate k, carrying capacity N,and time t measured in years. Adjust the model to account for each of the following situations. (a) One hundred fish are harvested each year. (b) One-third of the fish population is harvested annually. (c) The number of fish harvested each year is proportional to the square root of the number of fish in the lake.Suppose that the growth-rate parameter k = 0.3 and the carrying capacity N = 2500 in the logistic population model of Exercise 17. Suppose P(0) = 2500. (a) If 100 fish are harvested each year, what does the model predict for the long- term behavior of the fish population? In other words, what does a qualitative analysis of the model yield? (b) If one-third of the fish are harvested each year, what does the model predict for the long-term behavior of the fish population?The rhinoceros is now extremely rare. Suppose enough game preserve land is set aside so that there is sufficient room for many more rhinoceros territories than there are rhinoceroses. Consequently, there will be no danger of overcrowding. However, if the population is too small, fertile adults have difficulty finding each other when it is time to mate. Write a differential equation that models the rhinoceros population based on these assumptions. (Note that there is more than one reasonable model that fits these assumptions.)While it is difficult to imagine a time before cell phones, such a time did exist. The table below gives the number (in millions) of cell phone subscriptions in the United States from the U.S. census (see www.census.gov). Let s(t) be the number of cell phone subscriptions at times, measured in years since 1989. The relative growth rate of x(t) is its growth rate divided by the number of subscriptions. In other words, the relative growth rate is 1s(t)dsdt and it is often expressed as a percentage. (a) Estimate the relative growth rate of s(t) att = 1. That is, estimate the relative rate for the year 1990. Express this growth rate as a percentage. [Hint: The best estimate involves the number of cell phones at 1989 and 1991.] (b) In general, if a quantity grows exponentially, how does its relative growth rate change? (c) Also estimate the relative growth rates of s(t) for the years 1991—2007. (d) How long after 1989 was the number of subscriptions growing exponentially? (e) In general, if a quantity grows according to a logistic model, how does its relative growth rate change? (f) Using your results in part (c), calculate the carrying capacity for this model. [Hint: There is more than one way to do this calculation.]For the following predator-prey systems, identify which dependent variable,x or y, is the prey population and which is the predator population. Is the growth of the prey limited by any factors other than the number of predators? Do the predators have sources of food other than the prey? (Assume that the parameters ,,, and N are all positive.) (a) dxdt=x+xydydt=yxy (b) dxdt=xx2N+xydydt=y+xy In the following predator-prey population models,x represents the prey, and y represents the predators. (i) dxdt=5x3xydydt=2y+12xy (ii) dxdt=x8xydydt=2y+6xy (a) In which system does the prey reproduce more quickly when there are no predators (when y = 0) and equal numbers of prey? (b) In which system are the predators more successful at catching prey? In other words, if the number of predators and prey are equal for the two systems, in which system do the predators have a greater effect on the rate of change of the prey? (c) Which system requires more prey for the predators to achieve a given growth rate (assuming identical numbers of predators in both cases)?The following systems are models of the populations of pairs of species that either compete for resources (an increase in one species decreases the growth rate of the other) or cooperate (an increase in one species increases the growth rate of the other). For each system, identify the variables (independent and dependent) and the parameters (carrying capacity, measures of interaction between species. etc.) Do the species compete or cooperate? (Assume all parameters are positive.) (a) dxdt=xx2N+xydydt=y+xy (b) dxdt=x+xydydt=y+xy Bob. Glen. and Paul are once again sitting around enjoying their nice, cold glasses of iced cappucino when one of their students asks them to come up with solutions to the differential equation dydt=y+1t+1 After much discussion, Bob says y(t)= t, Glen says y(t) = 2t + 1, and Paul says y(t) = t2 2. (a) Who is right? (b) What solution should they have seen right away?Make up a differential equation of the form dydt=2yt+g(y) that has the function y(t) = e2tas a solution.Make up a differential equation of the form dy/dt =f(t,y) that has y(t) = et3 as a solution. (Try to come up with one whose right-hand side f(t, y) depends explicitly on both t and y.)In Section 1.1, we guessed solutions to the exponential growth model dP/dt = kP, where k is a constant (see page 6). Using the fact that this equation is separable, derive these solutions by separating variables.In Exercises 524, find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.) 5. dydt=(ty)2 In Exercises 524, find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.) 6. dydt=t4y In Exercises 524, find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.) 7. dydt=2y+1 In Exercises 524, find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.) 8. dydt=2y In Exercises 524, find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.) 9. dydt=ey In Exercises 524, find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.) 10. dxdt=1+x2 In Exercises 524, find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.) 11. dydt=2ty2+3y2 In Exercises 524, find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.) 12. dydt=ty In Exercises 524, find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.) 13. dydt=tt2y+y In Exercises 524, find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.) 14. dydt=ty3 In Exercises 524, find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.) 15. dydt=12y+1 In Exercises 524, find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.) 16. dydt=2y+1t In Exercises 524, find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.) 17. dydt=y(1y)In Exercises 524, find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.) 18. dydt=4t1+3y2 In Exercises 524, find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.) 19. dydt=t2v22v+t2 In Exercises 524, find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.) 20. dydt=1ty+t+y+1 In Exercises 524, find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.) 21. dydt=ety1+y2 In Exercises 524, find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.) 22. dydt=y24 In Exercises 524, find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.) 23. dwdt=wt In Exercises 524, find the general solution of the differential equation specified. (You may not be able to reach the ideal answer of an equation with only the dependent variable on the left and only the independent variable on the right, but get as far as you can.) 24. dydt=secy In Exercises 2538, solve the given initial-value problem. 25. dxdt=xt,y(0)=1/In Exercises 2538, solve the given initial-value problem. 26. dydt=ty,y(0)=3 In Exercises 2538, solve the given initial-value problem. 27. dydt=y2,y(0)=1/2 In Exercises 2538, solve the given initial-value problem. 28. dydt=t2y3,y(0)=1 In Exercises 2538, solve the given initial-value problem. 29. dydt=y2,y(0)=0 In Exercises 2538, solve the given initial-value problem. 30. dydt=tyt2y,y(0)=4 In Exercises 2538, solve the given initial-value problem. 31. dydt=2y+1,y(0)=3 32E 33E In Exercises 2538, solve the given initial-value problem. 34. dydt=1y2y,y(0)=2 In Exercises 2538, solve the given initial-value problem. 35. dydt=(y2+1)t,y(0)=1 In Exercises 2538, solve the given initial-value problem. 36. dydt=t2y+3,y(0)=1 In Exercises 2538, solve the given initial-value problem. 37. dydt=2ty2+3t2y2,y(1)=1 38E A 5-gallon bucket is full of pure water. Suppose we begin dumping salt into the bucket at a rate of 1/4 pounds per minute. Also, we open the spigot so that 1/2 gallons per minute leaves the bucket, and we add pure water to keep the bucket full. If the salt water solution is always well mixed, what is the amount of salt in the bucket after (a) 1 minute? (b) 10 minutes? (c)60 minutes? (d) 1000 minutes? (e) a very, very long time?Consider the following very simple model of blood cholesterol levels based on the fact that cholesterol is manufactured by the body for use in the construction of cell walls and is absorbed from foods containing cholesterol: Let C(t)be the amount (in milligrams per deciliter) of cholesterol in the blood of a particular person at time t(in days). Then dCdt=k1(NC)+k2E where N = the person’s natural cholesterol level. k1= production parameter. E = daily rate at which cholesterol is eaten, and k2= absorption parameter. (a) Suppose N = 200,k1= 0.1,k2= 0.1,E = 400, and C(0)= 150. What will the person’s cholesterol level be after 2 days on this diet? (b) With the initial conditions as above, what will the person’s cholesterol level be after 5 days on this diet? (c) What will the person’s cholesterol level be after a long time on this diet? (d) High levels of cholesterol in the blood are known to be a risk factor for heart disease. Suppose that, after a long time on the high cholesterol diet described above, the person goes on a very low cholesterol diet, so E changes to E = 100. (The initial cholesterol level at the starting time of this diet is the result of part (c).) What will the person’s cholesterol level be after 1 day on the new diet, after 5days on the new diet, and after a very long time on the new diet? (e) Suppose the person stays on the high cholesterol diet but takes drugs that block some of the uptake of cholesterol from food, so k2changes to k2= 0.075. With the cholesterol level from part (c), what will the person’s cholesterol level be after 1 day, after 5 days, and after a very long time?A cup of hot chocolate is initially 170o Fand is left in a room with an ambient temperature of 70o E Suppose that at time t = 0 it is cooling at a rate of 20oper minute. (a) Assume that Newton’s law of cooling applies: The rate of cooling is proportional to the difference between the current temperature and the ambient temperature. Write an initial-value problem that models the temperature of the hot chocolate. (b) How long does it take the hot chocolate to cool to a temperature of 110o F?Suppose you are having a dinner party for a large group of people, and you decide to make 2 gallons of chili. The recipe calls for 2 teaspoons of hot sauce per gallon, but you misread the instructions and put in 2 tablespoons of hot sauce per gallon. (Since each tablespoon is 3 teaspoons. you have put in 6 teaspoons per gallon. which is a total of 12 teaspoons of hot sauce in the chili.) You don’t want to throw the chili out because there isn’t much eke to cat (and some people like hot chili), so you serve the chili anyway. However, as each person takes some chili, you fill up the pot with beans and tomatoes without hot sauce until the concentration of hot sauce agrees with the recipe. Suppose the guests take I cup of chili per minute from the pot (there are 16 cups in a gallon), how long will it take to get the chili back to the recipe’s concentration of hot sauce? How many cups of chili will have been taken from the pot?43E In Exercises 1-6, sketch the slope fields for the differential equation as follows: (a) Pick a few points (t, y) with both 2t2 and 2y2 andplot the associated slope marks without the use of technology. (b) Use HPGSolver to check these individual slope marks. (c) Make a more detailed drawing of the slope field and then use HPGSolver to confirm your answer. For more details about HPGSolver and other programs that are pan of the DETools package, see the description of DEToo1s inside the front cover of this book. dydt=t2+t 2E 3E 4E 5E 6E In Exercises 710, a differential equation and its associated slope field are given. For each equation, (a) sketch a number of different solutions on the slope field, and (b) describe briefly the behavior of the solution with y(0) = 1/2 as t increases. You should first answer these exercises without using any technology, and then you should confirm your answer using HPGSolver. 7. dydt=3y(1y)8E 9E 10E Suppose we know that the function f(t, y) is continuous and that f(t. 3) = 1 for all t. (a) What does this information tell us about the slope field for the differential equation dy/dt = f(t, y)? (b) What can we conclude about solutions y(t)of dy/dt = f(t, y)?For example. if y(0) <3, can t as t increases?12E 13E 14E Consider the autonomous differential equation dSdt=S32S2+S (a) Make a rough sketch of the slope field without using any technology. (b) Using this drawing, sketch the graphs of the solutions S(t) with the initial conditionsS(0)= ½, S(1)= l/2,S(0)= 1.S(0)=3/2andS(0)= l/2. (c) Confirm your answer using HPGSolver.Eight differential equations and four slope fields are given below. Determine the equation that corresponds to each slope field and state briefly how you know your choice is correct. You should do this exercise without using technology. (i) dydt=y2+y (ii) dydt=y2y (iii) dydt=y3+y2 (iv) dydt=2t2 (v) dydt=ty+ty2 (vi) dydt=t2+t2y (vii) dydt=t+ty (viii) dydt=t22 17E 18E 19E 20E 21E 22E In Exercises 14, use EulersMethod to perform Euler’s method with the given step size t on the given initial-value problem over the time interval specified. Your answer should include a table of the approximate values of the dependent variable. It should also include a sketch of the graph of the approximate solution. 1. dydt=2y+1,y(0)=3,0t2,t=0.5 In Exercises 14, use EulersMethod to perform Euler’s method with the given step size t on the given initial-value problem over the time interval specified. Your answer should include a table of the approximate values of the dependent variable. It should also include a sketch of the graph of the approximate solution. 2. dydt=ty2,y(0)=1,0t1,t=0.25 3E In Exercises 14, use EulersMethod to perform Euler’s method with the given step size t on the given initial-value problem over the time interval specified. Your answer should include a table of the approximate values of the dependent variable. It should also include a sketch of the graph of the approximate solution. 4. dydt=siny,y(0)=1,0t2,t=0.5 In Exercises 510, use Euler’s method with the given step size t to approximate the solution to the given initial-value problem over the time interval specified. Your answer should include a table of the approximate values of the dependent variable. It should also include a sketch of the graph of the approximate solution. 5. dwdt=(3w)(w+1),w(0)=4,0t5,t=1.0 In Exercises 510, use Euler’s method with the given step size t to approximate the solution to the given initial-value problem over the time interval specified. Your answer should include a table of the approximate values of the dependent variable. It should also include a sketch of the graph of the approximate solution. 6. dwdt=(3w)(w+1),w(0)=0,0t5,t=0.5 7E 8E 9E 10E 11E 12E 13E 14E Consider the initial-value problem dy/dt= y ,y(0) = 1. Using Euler’s method, compute three different approximate solutions corresponding to t = 1.0, 0.5. and 0.25 over the interval 0t4 . Graph all three solutions. What predictions do you make about the actual solution to the initial-value problem?Consider the initial-value problem dy/dt= 2 y,y(0) = 1. Using Euler’s method, compute three different approximate solutions corresponding to t = 1.0, 0.5. and 0.25 over the interval 0t4 . Graph all three solutions. What predictions do you make about the actual solution to the initial-value problem? How do the graphs of these approximate solutions relate to the graph of the actual solution? Why?As we saw in Exercise 19 of Section 1.3, the spiking of a neuron can be modeled by the differential equation d/dt=1cos+(1+cos)I(t) , where I(t) is the input. Assume that I(t) is constantly equal to 0.1. Using Euler’s method with t = 0.1, graph the solution that solves the initial value (0)=1.0 over the interval 0t5 . When does the neuron spike?18E 19E 20E 21E In Exercises 1—4, we refer to a function f, but we do not provide its formula. However, we do assume that f satisfies the hypotheses of the Uniqueness Theorem in the entire ty-plane, and we do provide various solutions to the given differential equation. Finally, we specify an initial condition. Using the Uniqueness Theorem, what can you conclude about the solution to the equation with the given initial condition? 1. dydt=f(t,y)y1(t) for all t is a solution. initial condition y(0) = 1 2E In Exercises 1—4, we refer to a function f, but we do not provide its formula. However, we do assume that f satisfies the hypotheses of the Uniqueness Theorem in the entire ty-plane, and we do provide various solutions to the given differential equation. Finally, we specify an initial condition. Using the Uniqueness Theorem, what can you conclude about the solution to the equation with the given initial condition? 3. dydt=f(t,y) y1(t)=t+2 for all t is a solution. y2(t)=t2 for all t is a solution. initial condition y(0) = 1 In Exercises 1—4, we refer to a function f, but we do not provide its formula. However, we do assume that f satisfies the hypotheses of the Uniqueness Theorem in the entire ty-plane, and we do provide various solutions to the given differential equation. Finally, we specify an initial condition. Using the Uniqueness Theorem, what can you conclude about the solution to the equation with the given initial condition? 4. dydt=f(t,y) y1(t)=1 for all t is a solution. y2(t)=1+t2 for all t is a solution. initial condition y(0) = 1 5E In Exercises 5—8, an initial condition for the differential equation dydt=y(y1)(y3) is given. What does the Existence and Uniqueness Theorem say about the corresponding solution? 6. y(0)=0 7E In Exercises 5—8, an initial condition for the differential equation dydt=y(y1)(y3) is given. What does the Existence and Uniqueness Theorem say about the corresponding solution? 8. y(0)=1 (a) Show that y1(t)=t2 and y2(t)=t2+1 are solutions to dydt=y2+y+2yt2+2tt2t4 (b) Show that if y(t) is a solution to the differential equation in part (a) and if 0 y(0) 1, then t2y(t) 2 + 1 for all t. (c) Illustrate your answer using HPGSolver.Consider the differential equation dy/dt=2y (a) Show that the function y(t) = 0 for all t is an equilibrium solution. (b) Find all solutions. [Hint: Consider the cases y > 0 and y <0 separately. Then you need to define the solutions using language like “y(t) = ... when t0 and y(t) = ... when t>0.”] (c) Why doesn’t this differential equation contradict the Uniqueness Theorem? (d) What does HPGSolver do with this equation?Consider the differential equation dydt=yt2 (a) Show that the constant function y1(t) = 0 is a solution. (b)Show that there are infinitely many other functions that satisfy the differential equation, that agree with this solution when t0 ,but that are non zero when t> 0. [Hint: You need to define the solutions using language like “y(t) = ... when t0 and y(t) = ... when t>0.”] (c) Why doesn’t this example contradict the Uniqueness Theorem?(a) Show that y1(t)=1t1 and y2(t)=1t2 are solutions of dy/dt = y2. (b) What can you say about solutions of dy/dt = y2for which the initial condition y(0) satisfies the inequality 1 y(0) 1/2? [Hint: You could find the general solution, but what information can you get from your answer to part (a) alone?]13E In Exercises 13—16, an initial-value problem is given. (a) Find a formula for the solution. (b) State the domain of definition of the solution. (c) Describe what happens to the solution as it approaches the limits of its domain of definition. Why can’t the solution be extended for more lime? 14. dydt=1(y+1)(t2),y(0)=0 In Exercises 13—16, an initial-value problem is given. (a) Find a formula for the solution. (b) State the domain of definition of the solution. (c) Describe what happens to the solution as it approaches the limits of its domain of definition. Why can’t the solution be extended for more lime? 15. dydt=1( y+2)2,y(0)=1 In Exercises 13—16, an initial-value problem is given. (a) Find a formula for the solution. (b) State the domain of definition of the solution. (c) Describe what happens to the solution as it approaches the limits of its domain of definition. Why can’t the solution be extended for more lime? 16. dydt=ty2,y(1)=0 17E We have emphasized that the Uniqueness Theorem does not apply to every differential equation. There are hypotheses that must be verified before we can apply the theorem. However, there is a temptation to think that, since models of “real- world” problems must obviously have solutions, we don’t need to worry about the hypotheses of the Uniqueness Theorem when we are working with differential equations modeling the physical world. The following model illustrates the flaw in this assumption. Suppose we wish to study the formation of raindrops in the atmosphere. We make the reasonable assumption that raindrops arc approximately spherical. We also assume that the rate of growth of the volume of a raindrop is proportional to its surf ace area. Let r(t) be the radius of the raindrop at timet, s(t) be its surface area at time t, and v(t) be its volume at time t. From three-dimensional geometry, we know that s=4r2 and v=43r3 . (a) Show that the differential equation that models the volume of the raindrop und er these assumptions is dvdt=kv2/3 where k is a proportionality constant. (b) Why doesn’t this equation satisfy the hypotheses of the Uniqueness Theorem? (c) Give a physical interpretation of the fact that solutions to this equation with the initial condition v(0) = 0 are not unique. Does this model say anything about the way raindrops begin to form?In Exercises 112, sketch the phase lines for the given differential equation. Identify the equilibrium points as sinks, sources, or nodes. 1. dydt=3y(y2)In Exercises 112, sketch the phase lines for the given differential equation. Identify the equilibrium points as sinks, sources, or nodes. 2. dydt=y24y12 3E In Exercises 112, sketch the phase lines for the given differential equation. Identify the equilibrium points as sinks, sources, or nodes. 4. dwdt=wcosw 5E In Exercises 112, sketch the phase lines for the given differential equation. Identify the equilibrium points as sinks, sources, or nodes. 6. dydt=1y2 In Exercises 112, sketch the phase lines for the given differential equation. Identify the equilibrium points as sinks, sources, or nodes. 7. dvdt=v22v2 In Exercises 112, sketch the phase lines for the given differential equation. Identify the equilibrium points as sinks, sources, or nodes. 8. dwdt=3w312w2 9E In Exercises 112, sketch the phase lines for the given differential equation. Identify the equilibrium points as sinks, sources, or nodes. 10. dydt=tany 11E 12E In Exercises 1321, a differential equation and various initial conditions are specified. Sketch the graphs of the solutions satisfying these initial conditions. For each exercise, put all your graphs on one pair of axes. 13. Equation from Exercise 1; y(0)=1,y(2)=1,y(0)=3,y(0)=2 In Exercises 1321, a differential equation and various initial conditions are specified. Sketch the graphs of the solutions satisfying these initial conditions. For each exercise, put all your graphs on one pair of axes. 14. Equation from Exercise 2; y(0)=1,y(1)=0,y(0)=6,y(0)=5 In Exercises 1321, a differential equation and various initial conditions are specified. Sketch the graphs of the solutions satisfying these initial conditions. For each exercise, put all your graphs on one pair of axes. 15. Equation from Exercise 3; y(0)=0,y(1)=1,y(0)=/2,y(0)=16E ]In Exercises 1321, a differential equation and various initial conditions are specified. Sketch the graphs of the solutions satisfying these initial conditions. For each exercise, put all your graphs on one pair of axes. 17. Equation from Exercise 5; w(0)=3/2,w(0)=1,w(0)=2,w(0)=3 18E In Exercises 1321, a differential equation and various initial conditions are specified. Sketch the graphs of the solutions satisfying these initial conditions. For each exercise, put all your graphs on one pair of axes. 19. Equation from Exercise 7; v(0)=0,v(1)=1,v(0)=1 .20E 21E In Exercises 2227, describe the long-term behavior of the solution to the differential equation dydt=y24y+2with the given initial condition. 22. y(0)=1 23E In Exercises 2227, describe the long-term behavior of the solution to the differential equation dydt=y24y+2 with the given initial condition. 24. y(0)=2 In Exercises 2227, describe the long-term behavior of the solution to the differential equation dydt=y24y+2 with the given initial condition. 25. y(0)=4 In Exercises 2227, describe the long-term behavior of the solution to the differential equation dydt=y24y+2 with the given initial condition. 26. y(0)=4 27E 28E 29E In Exercises 2932, the graph of a function f(y) is given. Sketch the phase line for the autonomous differential equation dy/dt = f(y). 30.In Exercises 2932, the graph of a function f(y) is given. Sketch the phase line for the autonomous differential equation dy/dt = f(y). 31.In Exercises 2932, the graph of a function f(y) is given. Sketch the phase line for the autonomous differential equation dy/dt = f(y). 32.33E 34E 35E 36E Eight differential equations and four phase lines are given below. Determine the equation that corresponds to each phase line and state briefly how you know your choice is correct. (i) dydt=ycos(2,y) (ii) dydt=yy2 (iii) dydt=ysin(2,y) (iv) dydt=y3y2 (v) dydt=cos(2,y) (vi) dydt=y2+y (vii) dydt=ysin(2,y) (viii) dydt=y2y3 38E 39E Consider the Ermentrout-Kopell model for the spiking of a neuron ddt=1cos+(1+cos)I(t) introduced in Exercise 19 of Section 1.4. Let the input function I(t) be the function that is constantly s1/3. (a) Determine the equilibrium points for this input. (b) Classify these equilibria.41E 42E 43E 44E Let x(t) be the amount of time between two consecutive trolley cars as described above. We claim that a reasonable model for x(t) is dxdt=x Which term represents the rate of decrease of the time between the trains if no people are waiting, and which term represents the effect of the people waiting for the second train? (Justify your answer.) Should the parameters and be positive or negative?46E Use the model in Exercise 45 to predict what happens to x(t) ast increases. Include the effect of the initial value x(0). Is it possible for the trains to run at regular intervals? Given that there are always slight variations in the number of passengers waiting at each stop, is it likely that a regular interval can be maintained? Write two brief reports (of one or two paragraphs): (a) The first report is addressed to other students in the class (hence you may use technical language we use in class). (b) The second report is addressed to the Mayor of Boston.48E In Exercises 1-6, locate the bifurcation values for the one-parameter family and draw the phase lines for values of the parameter slightly smaller than, slightly larger than, and at the bifurcation values. 1. dydt=y2+In Exercises 1-6, locate the bifurcation values for the one-parameter family and draw the phase lines for values of the parameter slightly smaller than, slightly larger than, and at the bifurcation values. 2. dydt=y2+3y+a In Exercises 1-6, locate the bifurcation values for the one-parameter family and draw the phase lines for values of the parameter slightly smaller than, slightly larger than, and at the bifurcation values. 3. dydt=y2ay+1 In Exercises 1-6, locate the bifurcation values for the one-parameter family and draw the phase lines for values of the parameter slightly smaller than, slightly larger than, and at the bifurcation values. 4. dydt=y3+ay2 In Exercises 1-6, locate the bifurcation values for the one-parameter family and draw the phase lines for values of the parameter slightly smaller than, slightly larger than, and at the bifurcation values. 5. dydt=(y2)(y24)6E 7E 8E 9E 10E The graph to the right is the graph of a function f(y). Describe the bifurcations that occur in the one-parameter family dydt=f(y)+The graph to the right is the graph of a function g(y). Describe the bifurcations that occur in the one-parameter family dydt=g(y)+y [Hint: Note that the equilibria of this function occur at values of y where g(y)=y .]Six one-parameter families of different equations depending on the parameter A and four bifurcation diagrams are given below Determine the one-parameter family that corresponds to each bifurcation diagram, and state briefly how you know your choice is correct. (i) dydt=Ayy2 (ii) dydt=A+y2 (iii) dydt=Ayy3 (iv) dydt=Ay2 (v) dydt=y2A (vi) dydt=Ay+y2 Consider the Ermentrout-Kopell model for the spiking of a neuron ddt=1cos+(1+cos)I(t) introduced in Exercise 19 of Section 1.3. Suppose that the input function I(t) is a constant function, that is,I(t) = I where I is a constant. Describe the bifurcations that occur as the parameter I varies.15E Sketch the graph of a function g(y) such that the one-parameter family of differential equations dy/dt = g(y) + satisfies all of the following properties: • For all 4 , the differential equation has exactly two equilibria. • For all 4 , the equation has no equilibria. • For =0 , the equation has exactly four equilibria. [There are many possible functions g(y) that satisfy these conditions. Sketch just one graph.]Is it possible to find a continuous function f(y)such that the one-parameter family of differential equations dy/dt = f(y) + satisfies both of the following statements? • For =0 , the differential equation has exactly one equilibrium point and that equilibrium is a sink. • For =1 , the equation has exactly one equilibrium point and that equilibrium is a source. If so, sketch the graph of one such f(y). If not, why not?18E Consider the population model dPdt=2PP250 for a species of fish in a lake. Suppose it is decided that fishing will be allowed, but it is unclear how many fishing licenses should be issued. Suppose the average catch of a fisherman with a license is 3 fish per year (these arc hard fish to catch). (a) What is the largest number of licenses that can be issued lithe fish are to have a chance to survive in the lake? (b) Suppose the number of fishing licenses in part (a) is issued. What will happen to the fish population—that is, how does the behavior of the population depend on the initial population? (c) The simple population model above can be thought of as a model of an ideal fish population that is not subject to many of the environmental problems of an actual lake. For the actual fish population, there will be occasional changes in the population that were not considered when this model was constructed. For example, if the water level increases due to a heavy rainstorm, a few extra fish might be able to swim down a usually dry stream bed to reach the lake, or the extra water might wash toxic waste into the lake, killing a few fish. Given the possibility of unexpected perturbations of the population no included in the model, what do you think will happen to the actual fish population if we allow fishing at the level determined in pan (b)?20E 21E 22E 23E In Exercises 1-6, find the general solution of the equation specified. 1. dydt=4y+9et 2E In Exercises 1-6, find the general solution of the equation specified. 3. dydt=3y+4cos2t 4E In Exercises 1-6, find the general solution of the equation specified. 5. dy dt =3y4 e 3t In Exercises 1-6, find the general solution of the equation specified. 6. dydt=y2+4et/2 In Exercises 7-12, solve the given initial-value problem. 7. dydt+2y=et/3,y(0)=1 In Exercises 7-12, solve the given initial-value problem. 8. dydt2y=3e2t,y(0)=10 In Exercises 7-12, solve the given initial-value problem. 9. dydt+y=cos2t,y(0)=5 In Exercises 7-12, solve the given initial-value problem. 10. dydt+3y=cos2t,y(0)=1 In Exercises 7-12, solve the given initial-value problem. 11. dydt2y=7e2t,y(0)=3 In Exercises 7-12, solve the given initial-value problem. 12. dydt2y=7e2t,y(0)=3 Consider the nonhomogeneous linear equation dydt+2y=cos3t . To find a particular solution, it is pretty clear that our guess must contain a cosinefunction, but it is not so clear that the guess must also contain a sine function. (a) Guess yp(t)=cos3t and substitute this guess into the equation. Is there avalue of such that yp(t) is a solution? (b) Write a brief paragraph explaining why the proper guess for a particular solution is yp(t)=cos3t+sin3t .14E 15E 16E Consider the nonlinear differential equation dy/dt=y2 . (a) Show that y1(t)=1/(1t) isa solution. (b) Show that y2(t)=2/(1t) is not a solution. (C) Why don’t these two facts contradict the Linearity Principle?18E 19E Consider the nonhomogeneous linear equation dydt+2y=3t2+2t1 . In order to find the general solution, we must guess a particular solution yp(t) . Sincethe right-hand side is a quadratic polynomial, it is reasonable to guess a quadratic for yp(t) , so let yp(t)=at2+bt+c ,where a. b. and c are constants. Determine values for these constants so that yp(t) is a solution.In Exercises 21-24, find the general solution and the solution that satisfies the initialcondition y(0)=0 . 21. dydt+2y=t2+2t+1+e4t In Exercises 21-24, find the general solution and the solution that satisfies the initial condition y(0)=0 . 22. dydt+y=t3+sin3t In Exercises 21-24, find the general solution and the solution that satisfies the initial condition y(0)=0 . 23. dydt3y=2te4t In Exercises 21-24, find the general solution and the solution that satisfies the initial condition y(0)=0 . 24. dydt+y=cos2t+3sin2t+et 25E 26E In Exercises 25-28, give a brief qualitative description of the behavior of solutions. Note that we only give partial information about the functions in the differential equation, so your description must allow for various possibilities. Be sure to deal with initial conditions of different sizes and to discuss the long-term behavior of solutions. 27. dydt+y=b(t) , where b(t)3 for all t .28E A person initially places $1,000 in a savings account that pays interest at the rate of1.1 % per year compounded continuously. Suppose the person arranges for $20 perweek to be deposited automatically into the savings account. (a) Write a differential equation for P(t), the amount on deposit after t years (assume that “weekly deposits” is close enough to “continuous deposits” so thatwe may model the balance with a differential equation.) (b) Find the amount on deposit after 5 years.A student has saved $70,000 for her college tuition. When she starts college, she invests the money in a savings account that pays 1.5% interest per year, compoundedcontinuously. Suppose her college tuition is $30,000 per year and she arranges withthe college that the money will be deducted from her savings account in small payments. In other words, we assume that she is paying continuously. How long willshe be able to stay in school before she runs out of money?A college professor contributes $5,000 per year into her retirement fund by makingmany small deposits throughout the year. The fund grows at a rate of 7% per yearcompounded continuously. After 30 years, she retires and begins withdrawing fromher fund at a rate of $3000 per month. If she does not make any deposits after retirement, how long will the money last? [Hint: Solve this in two steps, before retirementand after retirement.]32E 33E 34E In Exercises 1-6, find the general solution of the differential equation specified. 1. dydt=yt+2 In Exercises 1-6, find the general solution of the differential equation specified. 2. dydt=3ty+t5 In Exercises 1-6, find the general solution of the differential equation specified. 3. dydt=y1+t+t2 In Exercises 1-6, find the general solution of the differential equation specified. 4. dydt=2ty+4et2 In Exercises 1-6, find the general solution of the differential equation specified. 5. dydt2t1+t2y=3 In Exercises 1-6, find the general solution of the differential equation specified. 6. dydt2ty=t3et In Exercises 7-12, solve the given initial-value problem. 7. dydt=y1+t+2,y(0)=3 In Exercises 7-12, solve the given initial-value problem. 8. dydt=11+ty+4t2+4t,y(1)=10 9E 10E 11E In Exercises 7-12, solve the given initial-value problem. 12. dydt3ty=2t3e2t,y(1)=0 In Exercises 13-18, the differential equation is linear, and in theory, we can find itsgeneral solution using the method of integrating factors. However, since this methodinvolves computing two integrals, in practice it is frequently impossible to reach a formula for the solution that is free of integrals. For these exercises, determine the generalsolution to the equation and express it with as few integrals as possible. 13. dydt=(sint)y+4 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E A 30-gallon tank initially contains 15 gallons of salt water containing 6 pounds ofsalt. Suppose salt water containing 1 pound of salt per gallon is pumped into the topof the tank at the rate of 2 gallons per minute, while a well-mixed Solution leavesthe bottom of the tank at a rate of 1 gallon per minute. How much salt is in the tankwhen the tank is full?A 400-gallon tank initially contains 200 gallons of water containing 2 parts per billion by weight of dioxin, an extremely potent carcinogen. Suppose water containing5 parts per billion of dioxin flows into the top of the tank at a rate of 4 gallons perminute. The water in the tank is kept well mixed, and 2 gallons per minute arc removed from the bottom of the tank. How much dioxin is in the tank when the tankis full?A 100-gallon tank initially contains loo gallons of sugar water at a concentration of 0.25 pounds of sugar per gallon. Suppose that sugar is added to the tank at a rate ofp pounds per minute, that sugar water is removed at a rate of 1 gallon per minute, and that the water in the tank is kept well mixed. (a) What value of p should we pick so that, when 5 gallons of sugar solution is leftin the tank, the concentration is 0.5 pounds of sugar per gallon? (b) Is it possible to choose p so that the last drop of water out of the bucket has aconcentration of 0.75 pounds of sugar per gallon?Suppose a 50-gallon tank contains a volume Vo of clean water at time t=0 . Attime t=0 , we begin dumping 2 gallons per minute of salt solution containing 0.25 pounds of salt per gallon into the tank. Also at time t=0 , we begin removing 1 gallon per minute of salt water from the tank. As usual, suppose the water inthe tank is well mixed so that the salt concentration at any given time is constant throughout the tank. (a) Set up the initial-value problem for the amount of salt in the tank. [Hint: The initial value of V0 will appear in the differential equation.] (b) What is your equation if V0=0 (the tank is initially empty)? Comment on thevalidity of the model in this situation. What will be the amount of salt in thetank at time t for this situation?Short answer exercises: Exercises 1-10 focus on the basic ideas, definitions, and vocabulary of this chapter. Their answers arc short (a single sentence or drawing), andyou should be able to do them with little or no computation. However, they vary indifficulty, so think carefully before you answer. 1. Give an example of a first-order differential equation that has the function y(t)=2t+3 as a solution.Short answer exercises: Exercises 1-10 focus on the basic ideas, definitions, and vocabulary of this chapter. Their answers arc short (a single sentence or drawing), and you should be able to do them with little or no computation. However, they vary in difficulty, so think carefully before you answer. 2. What is the general solution of the differential equation dy/dt=3y ?Short answer exercises: Exercises 1-10 focus on the basic ideas, definitions, and vocabulary of this chapter. Their answers arc short (a single sentence or drawing), and you should be able to do them with little or no computation. However, they vary in difficulty, so think carefully before you answer. 3. Find all equilibrium solutions for the differential equation dy/dt=t2(t2+1) .4RE Short answer exercises: Exercises 1-10 focus on the basic ideas, definitions, and vocabulary of this chapter. Their answers arc short (a single sentence or drawing), and you should be able to do them with little or no computation. However, they vary in difficulty, so think carefully before you answer. 5. Find all of the equilibrium solutions for the differential equation dydt=(t24)(1+y)ey(t1)(3y) .Short answer exercises: Exercises 1-10 focus on the basic ideas, definitions, and vocabulary of this chapter. Their answers arc short (a single sentence or drawing), and you should be able to do them with little or no computation. However, they vary in difficulty, so think carefully before you answer. 6. Sketch the phase line for the autonomous equation dy/dt=sin2y .7RE 8RE 9RE 10RE 11RE 12RE 13RE 14RE True-false: For Exercises 11-20, determine if the statement is true or false. If it is true, explain why. If it is false, provide a counterexample or an explanation. 15. Every separable differential equation is a homogeneous linear equation.16RE True-false: For Exercises 11-20, determine if the statement is true or false. If it is true, explain why. If it is false, provide a counterexample or an explanation. 17. The solution of dy/dt=(y3)(sinysint+cost+1) with y(0)=4 satisfies y(t)3 for all t.18RE 19RE 20RE 21RE 22RE 23RE 24RE 25RE 26RE 27RE 28RE In Exercises 21-29, (a) specify if the given equation is autonomous, linear and homogeneous, linear and nonhomogeneous, and/or separable, and (b) find its general solution. 29. dydt=3y+e2t+t2 30RE 31RE 32RE 33RE 34RE In Exercises 30-39, (a) specify if the given equation is autonomous, linear and homogeneous, linear and nonhomogeneous, and/or separable, and (b) solve the initial-value problem. 35. dydt=2ty+3tet2,y(0)=1 36RE 37RE 38RE 39RE 40RE 41RE

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