True-False Review
For Questions (a)-(i), decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem in fact from the text. If false, provide an example, illustration, or brief explanation of why the statement is false.
(b) The function
Want to see the full answer?
Check out a sample textbook solutionChapter 1 Solutions
Differential Equations and Linear Algebra (4th Edition)
- Turbo-Charged Engine Versus Standard Engine In tests conducted by Auto Test Magazine on two identical models of the Phoenix Elite—one equipped with a standard engine and the other with a turbo-charger—it was found that the acceleration of the former is given by a = f(t) = 4 + 0.8t (0 ≤ t ≤ 12) ft/sec/sec, t sec after starting from rest at full throttle, whereas the acceleration of the latter is given by a = g(t) = 4 + 1.2t + 0.03t2 (0 ≤ t ≤ 12) ft/sec/sec. How much faster is the turbo-charged model moving than the model with the standard engine at the end of a 9-sec test run at full throttle? ________ft/secarrow_forwardHomogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degree n if f(ʎx, ʎy) = ʎnf(x,y) Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v = x/y to have the GS in terms of x and y have C which is arbitrary constant Solve the DE. (Homogeneous DE) 1. xdx + sin2 (y/x) [ydx - xdy = 0]arrow_forwardHomogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degree n if f(ʎx, ʎy) = ʎnf(x,y) Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v = x/y to have the GS in terms of x and y have C which is arbitrary constant Solve the DE. (Homogeneous DE) 1. x2y' = 4x2 + 7xy + 2y2arrow_forward
- Homogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degree n if f(ʎx, ʎy) = ʎnf(x,y) Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v = x/y to have the GS in terms of x and y have C which is arbitrary constant Solve the DE. (Homogeneous DE) 1. [y - (x2+y2)1/2]dx - xdy = 0arrow_forwardHomogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degree n if f(ʎx, ʎy) = ʎnf(x,y) Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v = x/y to have the GS in terms of x and y have C which is arbitrary constant Solve the DE. (Homogeneous DE) 1. (x3 + y3) dx + 3xy2dy = 0arrow_forwardHomogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degree n if f(ʎx, ʎy) = ʎnf(x,y) Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v = x/y to have the GS in terms of x and y have C which is arbitrary constant Solve the DE. (Homogeneous DE) 1. (y2-x2)dx + xydy = 0arrow_forward
- Homogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degree n if f(ʎx, ʎy) = ʎnf(x,y) Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v = x/y to have the GS in terms of x and y have C which is arbitrary constant Solve the DE. (Homogeneous DE) 1. [xcos2(y/x) - y]dx + xdy = 0arrow_forwardHomogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degree n if f(ʎx, ʎy) = ʎnf(x,y) Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v = x/y to have the GS in terms of x and y have C which is arbitrary constant Solve the DE. (Homogeneous DE) 1. ydx = [x + (y2 - x2)1/2]dyarrow_forwardHomogeneous DE M(x,y)dx + N(x,y)dy = 0 is Homogeneous if both M and N are homogeneous functions of the same order. A function f(x,y) is called homogeneous of degree n if f(ʎx, ʎy) = ʎnf(x,y) Solution Steps: Objective: To reduce into a Variable Separable Form a. Replace y by ux or x by vy if y = ux, dy = udx + xdu . Use y=ux if N is simpler in form. if x = vy, dx = vdy + ydv . Use x=y if M is simpler in form b. Simplify the resulting equation c. Separate the variables d. Integrate both sides of the equation to get the General Solution e. Substitute u = y/x or v = x/y to have the GS in terms of x and y have C which is arbitrary constant Solve the DE. (Homogeneous DE) 1. xydx - (x + 2y)2dy - 0arrow_forward
- Doing practice problems from the textbook. What are the critical points of f(x,y) = (e^x)(sin(y)-1)arrow_forwardJoint PDF Hard Problem: If f_X,Y (x,y) = 6e^-(2x + 3y) for x, y >=0 then what is P[X+Y <=1]?arrow_forwardDifferential Applications y = f (x) A company produces x units of goods at a cost (5x2 - 10x + 30) in thousands of rupiah for each unit. If the goods are sold out at a price of IDR 50,000.00 per unit, then the maximum profit the company will get is ...arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning