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All Textbook Solutions for A First Course in Differential Equations with Modeling Applications (MindTap Course List)
19RESawing Wood A long uniform piece of wood (cross sections are the same) is cut through perpendicular to its length by a vertical saw blade. See Figure 3.R.6. If the friction between the sides of the saw blade and the wood through which the blade passes is ignored, then it can be assumed that the rate at which the saw blade moves through the piece of wood is inversely proportional to the width of the wood in contact with its cutting edge. As the blade advances through the wood (moving, say, left to right) the width of a cross section changes as a nonnegative continuous function w. If a cross section of the wood is described as a region in the xy-plane defined over an interval [a, b] then, as shown in Figure 3.R.6(c), the position x of the saw blade is a function of time t and the vertical cut made by the blade can be represented by a vertical line segment. The length of this vertical line is the width w(x) of the wood at that point. Thus the position x(t) of the saw blade and the rate dx/dt at which it moves to the right are related to w(x) by w(x)dxdt=k,x(0)=a. FIGURE 3.R.6 Sawing a log in Problem 20 Here k represents the number of square units of the material removed by the saw blade per unit time. (a) Suppose the saw is computerized and can be programmed so that k = 1. Find an implicit solution of the foregoing initial-value problem when the piece of wood is a circular log. Assume a cross section is a circle of radius 2 centered at (0, 0). [Hint: To save time see formula 41 in the table of integrals given on the right inside page of the front cover.] (b) Solve the implicit solution obtained in part (b) for time t as a function of x. Graph the function t(x). With the aid of the graph, approximate the time that it takes the saw to cut through this piece of wood. Then find the exact value of this time.Solve the initial-value problem in Problem 20 when a cross section of a uniform piece of wood is the triangular region given in Figure 3.R.7. Assume again that k = 1. How long does it take to cut through this piece of wood? FIGURE 3.R.7 Triangular cross section in Problem 2122REIn Problems 14 the given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem. 1. y = c1ex + c2ex, (, ); y y = 0, y(0) = 0, y(0) = 1In Problems 14 the given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem. 2. y=c1e4x+c2ex,(,);y3y4y=0,y(0)=1,y(0)=2In Problems 14 the given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem. 3. y=c1x+c2xlnx,(0,);x2yxy+y=0,y(1)=3,y(1)=1In Problems 14 the given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem. 4. y=c1+c2cosx+c3sinx,(,);y+y=0,y()=0,y()=2,y()=1Given that y=c1+c2x2 is a two-parameter family of solutions of xyy=0 on the interval (,), show that constants c1 and c2 cannot be found so that a member of the family satisfies the initial conditions y(0) = 0, y(0) = 1. Explain why this does not violate Theorem 4.1.1.Find two members of the family of solutions in Problem 5 that satisfy the initial conditions y(0)=0,y(0)=0.Given that x(t) = c1 cos t + c2 sin t is the general solution of x + 2x = 0 on the interval (, ), show that a solution satisfying the initial conditions x(0) = x0, x(0) = x1 is given by x(t)=x0cost+x1sint.Use the general solution of x + 2x = 0 given in Problem 7 to show that a solution satisfying the initial conditions x(t0) = x0, x(t0) = x1 is the solution given in Problem 7 shifted by an amount t0: x(t)=x0cos(tt0)+x1sin(tt0). 7. Given that x(t) = c1 cos t + c2 sin t is the general solution of x + 2x = 0 on the interval (, ), show that a solution satisfying the initial conditions x(0) = x0, x(0) = x1 is given by x(t)=x0cost+x1sint.In Problems 9 and 10 find an interval centered about x = 0 for which the given initial-value problem has a unique solution. 9. (x 2)y + 3y = x, y(0) = 0, y(0) = 1In Problems 9 and 10 find an interval centered about x = 0 for which the given initial-value problem has a unique solution. 10. y + (tan x)y = ex, y(0) = 1, y(0) = 0(a) Use the family in Problem 1 to find a solution of y y = 0 that satisfies the boundary conditions y(0) = 0, y(1) = 1. (b) The DE in part (a) has the alternative general solution y = c3 cosh x + c4 sinh x on (, ). Use this family to find a solution that satisfies the boundary conditions in part (a). (c) Show that the solutions in parts (a) and (b) are equivalent 1. y=c1ex+c2ex,(,);yy=0,y(0)=0,y(0)=1Use the family in Problem 5 to find a solution of xyy=0 that satisfies the boundary conditions y(0)=1, y(1)=6.In Problems 13 and 14 the given two-parameter family is a solution of the indicated differential equation on the interval (, ). Determine whether a member of the family can be found that satisfies the boundary conditions. 13. y=c1excosx+c2exsinx;y2y+2y=0 (a)y(0)=1,y()=0(b)y(0)=1,y()=1(c)y(0)=1,y(/2)=1(d)y(0)=0,y()=0.In Problems 13 and 14 the given two-parameter family is a solution of the indicated differential equation on the interval (, ). Determine whether a member of the family can be found that satisfies the boundary conditions. 14. y=c1x2+c2x4+3;x2y5xy+8y=24 (a)y(1)=0,y(1)=4(b)y(0)=1,y(1)=2(c)y(0)=3,y(1)=0(d)y(1)=3,y(2)=15In Problems 1522 determine whether the given set of functions is linearly independent on the interval (, ). 15. f1(x) = x, f2(x) = x2, f3(x) = 4x 3x2In Problems 1522 determine whether the given set of functions is linearly independent on the interval (, ). 16. f1(x) = 0, f2(x) = x, f3(x) = exIn Problems 1522 determine whether the given set of functions is linearly independent on the interval (, ). 17. f1(x) = 5, f2(x) = cos2x, f3(x) = sin2xIn Problems 1522 determine whether the given set of functions is linearly independent on the interval (, ). 18. f1(x) = cos 2x, f2(x) = 1, f3(x) = cos2xIn Problems 1522 determine whether the given set of functions is linearly independent on the interval (, ). 19. f1(x) = x, f2(x) = x 1, f3(x) = x + 3In Problems 1522 determine whether the given set of functions is linearly independent on the interval (, ). 20. f1(x) = 2 + x, f2(x) = 2 + |x|In Problems 1522 determine whether the given set of functions is linearly independent on the interval (, ). 21. f1(x) = 1 + x, f2(x) = x, f3(x) = x2In Problems 1522 determine whether the given set of functions is linearly independent on the interval (, ). 22. f1(x) = ex, f2(x) = ex, f3(x) = sinh xIn Problems 2330 verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. 23. yy12y=0;e3x,e4x,(,)In Problems 2330 verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. 24. y4y=0;cosh2x,sinh2x,(,)In Problems 2330 verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. 24. y2y+5y=0;excos2x,exsin2x,(,)In Problems 2330 verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. 26. 4y4y+y=0;ex/2,xex/2,(,)In Problems 2330 verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. 27. x2y6xy+12y=0;x3,x4(0,)In Problems 2330 verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. 28. x2y+xy+y=0;cos(lnx),sin(lnx),(0,)In Problems 2330 verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. 29. x3x+6x2y+4xy4y=0;x,x2,x2lnx,(0,)In Problems 2330 verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution. 30. y(4)+y=0;1,x,cosx,sinx,(,)In Problems 3134 verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval. 31. y7y+10y=24ex;y=c1e2x+c2e5x+6ex,(,)In Problems 3134 verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval. 32. y+y=secx;y=c1cosx+c2sinx+xsinx+(cosx)ln(cosx),(/2,/2)In Problems 3134 verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval. 33. y+4y+4y=2e2x+4x12;y=c1e2x+c2xe2x+x2e2x+x2,(,)In Problems 3134 verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval. 33. 2x2y+5xy+y=x2x;y=c1x1/2+c2x1+115x216x,(0,)(a) Verify that yp1=3e2x and yp2=x2+3xare, respectively, particular solutions of y6y+5y=9e2x and y6y+5y=5x2+3x16 (b) Use part (a) to find particular solutions of y6y+5y=5x2+3x169e2x and y6y+5y=10x26x+32+e2x(a) By inspection find a particular solution of y+2y=10. (b) By inspection find a particular solution of y+2y=4x. (c) Find a particular solution of y + 2y = 4x + 10. (d) Find a particular solution of y + 2y = 8x + 5.37E38E(a) Verify that y1 = x3 and y2 = |x|3 are linearly independent solutions of the differential equation x2y 4xy + 6y = 0 on the interval (, ). (b) For the functions y1 and y2 in part (a), show that W(y1, y2) = 0 for every real number x. Does this result violate Theorem 4.1.3? Explain. (c) Verify that Y1 = x3 and Y2 = x2 are also linearly independent solutions of the differential equation in part (a) on the interval (, ). (d) Besides the functions y1, y2, Y1, and Y2 in parts (a) and (c), find a solution of the differential equation that satisfies y(0) = 0, y(0) = 0. (e) By the superposition principle, Theorem 4.1.2, both linear combinations y = c1y1 + c2y2 and Y = c1Y1 + C2Y2 are solutions of the differential equation. Discuss whether one, both, or neither of the linear combinations is a general solution of the differential equation on the interval (, ).Is the set of functions f1(x) = ex+2, f2(x) = ex3 linearly dependent or linearly independent on (, )? Discuss.41ESuppose that y1, y2, , yk are k nontrivial solutions of a homogeneous linear nth-order differential equation with constant coefficients and that k = n + 1. Is the set of solutions y1, y2, , yk linearly dependent or linearly independent on (, )? Discuss.In Problems 116 the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x). 1. y 4y + 4y = 0; y1 = e2xIn Problems 116 the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x). 2. y + 2y + y = 0; y1 = xexIn Problems 116 the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x). 3. y + 16y = 0; y1 = cos 4xIn Problems 116 the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x). 4. y + 9y = 0; y1 = sin 3xIn Problems 116 the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x). 5. y y = 0; y1 = cosh xIn Problems 116 the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x). 6. y 25y = 0; y1 = e5xIn Problems 116 the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x). 7. 9y 12y + 4y = 0; y1 = e2x/3In Problems 116 the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x). 8. 6y + y y = 0; y1 = ex/3In Problems 116 the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x). 9. x2y 7xy + 16y = 0; y1 = x4In Problems 116 the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x). 10. x2y + 2xy 6y = 0; y1 = x2In Problems 116 the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x). 11. xy + y = 0; y1 = ln xIn Problems 116 the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x). 12. 4x2y + y = 0; y1 = x1/2 ln xIn Problems 116 the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x). 13. x2y xy + 2y = 0; y1 = x sin(ln x)In Problems 116 the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x). 14. x2y 3xy + 5y = 0; y1 = x2 cos(ln x)In Problems 116 the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x). 15. (1 2x x2)y + 2(1 + x)y 2y = 0; y1 = x + 1In Problems 116 the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x). 16. (1 x2)y + 2xy = 0; y1 = 1In Problems 1720 the indicated function y1(x) is a solution of the associated homogeneous equation. Use the method of reduction of order to find a second solution y2(x) of the homogeneous equation and a particular solution yp(x) of the given nonhomogeneous equation. 17. y 4y = 2; y1 = e2xIn Problems 1720 the indicated function y1(x) is a solution of the associated homogeneous equation. Use the method of reduction of order to find a second solution y2(x) of the homogeneous equation and a particular solution yp(x) of the given nonhomogeneous equation. 18. y + y = 1; y1 = 1In Problems 1720 the indicated function y1(x) is a solution of the associated homogeneous equation. Use the method of reduction of order to find a second solution y2(x) of the homogeneous equation and a particular solution yp(x) of the given nonhomogeneous equation. 19. y 3y + 2y = 5e3x; y1 = exIn Problems 1720 the indicated function y1(x) is a solution of the associated homogeneous equation. Use the method of reduction of order to find a second solution y2(x) of the homogeneous equation and a particular solution yp(x) of the given nonhomogeneous equation. 20. y 4y + 3y = x; y1 = ex21EIn Problems 21 and 22 the indicated function y1(x) is a solution of the given differential equation. Use formula (5) to find a second solution y2(x) expressed in terms of an integral-defined function. See (iii) in the Remarks. 22. 2xy (2x + 1)y + y = 0; y1 = ex23EVerify that y1(x) = x is a solution of xy" xy + y = 0. Use reduction of order to find a second solution y2(x) in the form of an infinite series. Conjecture an interval of definition for y2(x).In Problems 114 find the general solution of the given second-order differential equation. 1. 4y + y = 0In Problems 114 find the general solution of the given second-order differential equation. 2. y 36y = 0In Problems 114 find the general solution of the given second-order differential equation. 3. y y 6y = 0In Problems 114 find the general solution of the given second-order differential equation. 4. y 3y + 2y = 0In Problems 114 find the general solution of the given second-order differential equation. 5. y + 8y + 16y = 0In Problems 114 find the general solution of the given second-order differential equation. 5. y 10y + 25y = 0In Problems 114 find the general solution of the given second-order differential equation. 7. 12y 5y 2y = 0In Problems 114 find the general solution of the given second-order differential equation. 8. y + 4y y = 0In Problems 114 find the general solution of the given second-order differential equation. 9. y + 9y = 0In Problems 114 find the general solution of the given second-order differential equation. 10. 3y + y = 0In Problems 114 find the general solution of the given second-order differential equation. 11. y 4y + 5y = 0In Problems 114 find the general solution of the given second-order differential equation. 12. 2y + 2y + y = 0In Problems 114 find the general solution of the given second-order differential equation. 13. 3y + 2y + y = 0In Problems 114 find the general solution of the given second-order differential equation. 14. 2y 3y + 4y = 0In Problems 1528 find the general solution of the given higher-order differential equation. 15. y 4y 5y = 0In Problems 1528 find the general solution of the given higher-order differential equation. 16. y y = 0In Problems 1528 find the general solution of the given higher-order differential equation. 17. y 5y + 3y + 9y = 0In Problems 1528 find the general solution of the given higher-order differential equation. 18. y + 3y 4y 12y = 0In Problems 1528 find the general solution of the given higher-order differential equation. 19. d3udt3+d2udt22u=0In Problems 1528 find the general solution of the given higher-order differential equation. 20. d3xdt3d2xdt24x=0In Problems 1528 find the general solution of the given higher-order differential equation. 21. dPdt=p(1p);P=c1et1+c1etIn Problems 1528 find the general solution of the given higher-order differential equation. 22. y 6y + 12y 8y = 0In Problems 1528 find the general solution of the given higher-order differential equation. 23. y(4) + y + y = 0In Problems 1528 find the general solution of the given higher-order differential equation. 24. y(4) 2y + y = 0In Problems 1528 find the general solution of the given higher-order differential equation. 25. 16d4ydx4+24d2ydx+9y=0In Problems 1528 find the general solution of the given higher-order differential equation. 26. d4ydx47d2ydx218y=0In Problems 1528 find the general solution of the given higher-order differential equation. 27. d5udr5+5d4udr42d3udr310d2udr2+dudr+5u=028EIn Problems 2936 solve the given initial-value problem. 29. y+16y=0,y(0)=2,y(0)=2In Problems 2936 solve the given initial-value problem. 30. d2yd2+y=0,y(/3)=0,y(/3)=2In Problems 2936 solve the given initial-value problem. 31. d2ydt24dydt5t=0,y(1)=0,y(1)=2In Problems 2936 solve the given initial-value problem. 32. 4y4y3y=0,y(0)=1,y(0)=5In Problems 2936 solve the given initial-value problem. 33. y+y+2y=0,y(0)=y(0)=034EIn Problems 2936 solve the given initial-value problem. 35. y+12y+36y=0,y(0)=0,y(0)=1,y(0)=736EIn Problems 3740 solve the given boundary-value problem. 37. y10y+25y=0,y(0)=1,y(1)=038EIn Problems 3740 solve the given boundary-value problem. 39. y+y=0,y(0)=0,y(/2)=040E41E42EIn Problems 4348 each figure represents the graph of a particular solution of one of the following differential equations: (a) y 3y 4y = 0 (b) y + 4y = 0 (c) y + 2y + y = 0 (d) y + y = 0 (e) y + 2y + 2y = 0 (f) y 3y + 2y = 0 Match a solution curve with one of the differential equations. Explain your reasoning. 43. Figure 4.3.2 Graph for Problem 4344EIn Problems 4348 each figure represents the graph of a particular solution of one of the following differential equations: (a) y 3y 4y = 0 (b) y + 4y = 0 (c) y + 2y + y = 0 (d) y + y = 0 (e) y + 2y + 2y = 0 (f) y 3y + 2y = 0 Match a solution curve with one of the differential equations. Explain your reasoning. 45. Graph for Problem 4546EIn Problems 4348 each figure represents the graph of a particular solution of one of the following differential equations: (a) y 3y 4y = 0 (b) y + 4y = 0 (c) y + 2y + y = 0 (d) y + y = 0 (e) y + 2y + 2y = 0 (f) y 3y + 2y = 0 Match a solution curve with one of the differential equations. Explain your reasoning. 47. Figure 4.3.6 Graph for Problem 4748EIn Problems 4958 find a homogeneous linear differential equation with constant coefficients whose general solution is given. 49. y=c1ex+c2e5xIn Problems 4958 find a homogeneous linear differential equation with constant coefficients whose general solution is given. 50. y = c1e4x + c2e3xIn Problems 4958 find a homogeneous linear differential equation with constant coefficients whose general solution is given. 51. y = c1 + c2e2x52EIn Problems 4958 find a homogeneous linear differential equation with constant coefficients whose general solution is given. 53. y = c1 cos 3x + c2 sin 3xIn Problems 4958 find a homogeneous linear differential equation with constant coefficients whose general solution is given. 54. y = c1 cosh 7x + c2 sinh 7xIn Problems 4958 find a homogeneous linear differential equation with constant coefficients whose general solution is given. 55. y = c1ex cos x + c2ex sin xIn Problems 4958 find a homogeneous linear differential equation with constant coefficients whose general solution is given. 56. y = c1 + c2e2x cos 5x + c3e2x sin 5 x57EIn Problems 4958 find a homogeneous linear differential equation with constant coefficients whose general solution is given. 58. y = c1 cos x + c2 sin x + c3 cos 2 x + c4 sin 2 xTwo roots of a cubic auxiliary equation with real coefficients are m1=12 and m2 = 3 + i. What is the corresponding homogeneous linear differential equation? Discuss: Is your answer unique?60E61E62E63EConsider the boundary-value problem y+y=0, y(0) = 0, y(/2) = 0. Discuss: Is it possible to determine real values of so that the problem possesses (a) trivial solutions? (b) nontrivial solutions?In Problems 126 solve the given differential equation by undetermined coefficients. 1. y+3y+2y=6In Problems 126 solve the given differential equation by undetermined coefficients. 2. 4y + 9y = 15In Problems 126 solve the given differential equation by undetermined coefficients. 3. y10y+25y=30x+3In Problems 126 solve the given differential equation by undetermined coefficients. 4. y+y6y=2xIn Problems 126 solve the given differential equation by undetermined coefficients. 5. 14y+y+y=x22xIn Problems 126 solve the given differential equation by undetermined coefficients. 6. y 8y + 20y = 100x2 26xexIn Problems 126 solve the given differential equation by undetermined coefficients. 7. y + 3y = 48x2e3xIn Problems 126 solve the given differential equation by undetermined coefficients. 8. 4y 4y 3y = cos 2xIn Problems 126 solve the given differential equation by undetermined coefficients. 9. y y = 3In Problems 126 solve the given differential equation by undetermined coefficients. 10. y + 2y = 2x + 5 e2xIn Problems 126 solve the given differential equation by undetermined coefficients. 11. yy+14y=3+ex/2In Problems 126 solve the given differential equation by undetermined coefficients. 12. y 16y = 2e4xIn Problems 126 solve the given differential equation by undetermined coefficients. 13. y + 4y = 3 sin 2xIn Problems 126 solve the given differential equation by undetermined coefficients. 14. y 4y = (x2 3) sin 2xIn Problems 126 solve the given differential equation by undetermined coefficients. 15. y + y = 2x sin xIn Problems 126 solve the given differential equation by undetermined coefficients. 16. y 5y = 2x3 4x2 x + 6In Problems 126 solve the given differential equation by undetermined coefficients. 17. y 2y + 5y = ex cos 2xIn Problems 126 solve the given differential equation by undetermined coefficients. 18. y 2y + 2y = e2x(cos x 3 sin x)In Problems 126 solve the given differential equation by undetermined coefficients. 19. y + 2y + y = sin x + 3 cos 2xIn Problems 126 solve the given differential equation by undetermined coefficients. 20. y + 2y 24y = 16 (x + 2)e4xIn Problems 126 solve the given differential equation by undetermined coefficients. 21. y 6y = 3 cos xIn Problems 126 solve the given differential equation by undetermined coefficients. 22. y 2y 4y + 8y = 6xe2xIn Problems 126 solve the given differential equation by undetermined coefficients. 23. y 3y + 3y y = x 4exIn Problems 126 solve the given differential equation by undetermined coefficients. 24. y y 4y + 4y = 5 ex + e2xIn Problems 126 solve the given differential equation by undetermined coefficients. 25. y(4) + 2y + y = (x 1)2In Problems 126 solve the given differential equation by undetermined coefficients. 26. y(4) y = 4x + 2xexIn Problems 2736 solve the given initial-value problem. 27. y + 4y = 2, y(/8) = 12, y(/8) = 2In Problems 2736 solve the given initial-value problem. 28. 2y + 3y 2y = 14x2 4x 11, y(0) = 0, y(0) = 0In Problems 2736 solve the given initial-value problem. 29. 5y + y = 6x, y(0) = 0, y(0) = 10In Problems 2736 solve the given initial-value problem. 30. y + 4y + 4y = (3 + x)e2x, y(0) = 2, y(0) = 5In Problems 2736 solve the given initial-value problem. 31. y + 4y + 5y = 35e4x, y(0) = 3, y(0) = 1In Problems 2736 solve the given initial-value problem. 32. y y = cosh x, y(0) = 2, y(0) = 12In Problems 2736 solve the given initial-value problem. 33. d2xdt2+2x = F0 sin t, x(0) = 0, x(0) = 0In Problems 2736 solve the given initial-value problem. 34. d2xdt2+2x = F0 sin t, x(0) = 0, x(0) = 0In Problems 2736 solve the given initial-value problem. 35. y 2y + y = 2 24ex + 40e5x, y(0) = 12, y(0) = 52, y(0) = 92In Problems 2736 solve the given initial-value problem. 36. y + 8y = 2x 5 + 8e2x, y(0) = 5, y(0) = 3, y(0) = 4In Problems 3740 solve the given boundary-value problem. 37. y + y = x2 + 1, y(0) = 5, y(1) = 0In Problems 3740 solve the given boundary-value problem. 38. y 2y + 2y = 2x 2, y(0) = 0, y() =In Problems 3740 solve the given boundary-value problem. 39. y + 3y = 6x, y(0) = 0, y(1) + y(1) = 0In Problems 3740 solve the given boundary-value problem. 40. y + 3y = 6x, y(0) + y(0) = 0, y(1) = 0In Problems 41 and 42 solve the given initial-value problem in which the input function g(x) is discontinuous. [Hint: Solve each problem on two intervals, and then find a solution so that y and y are continuous at x = /2 (Problem 41) and at x = (Problem 42).] 41. y + 4y = g(x), y(0) = 1, y(0) = 2, where g(x)={sinx,0x/20,x/2In Problems 41 and 42 solve the given initial-value problem in which the input function g(x) is discontinuous. [Hint: Solve each problem on two intervals, and then find a solution so that y and y are continuous at x = /2 (Problem 41) and at x = (Problem 42).] 42. y 2y + 10y = g(x), y(0) = 0, y(0) = 0, where g(x)={20,0x0,xConsider the differential equation ay + by + cy = ekx, where a, b, c, and k are constants. The auxiliary equation of the associated homogeneous equation is am2 + bm + c = 0. (a) If k is not a root of the auxiliary equation, show that we can find a particular solution of the form yp = Aekx, where A = 1 /(ak2 + bk + c). (b) If k is a root of the auxiliary equation of multiplicity one, show that we can find a particular solution of the form yp = Axekx, where A = 1/(2ak + b). Explain how we know that k b/(2a). (c) If k is a root of the auxiliary equation of multiplicity two, show that we can find a particular solution of the form y = Ax2ekx, where A = 1 /(2a).44EIn Problems 4548 without solving, match a solution curve of y + y = f(x) shown in the figure with one of the following functions: (i) f(x) = 1, (ii) f(x) = ex, (iii) f(x) = ex, (iv) f(x) = sin 2x, (v) f(x) = ex sin x, (vi) f (x) = sin x. Briefly discuss your reasoning. 45. FIGURE 4.4.1 Graph for Problem 45In Problems 4548 without solving, match a solution curve of y + y = f(x) shown in the figure with one of the following functions: (i) f(x) = 1, (ii) f(x) = ex, (iii) f(x) = ex, (iv) f(x) = sin 2x, (v) f(x) = ex sin x, (vi) f(x) = sin x. 46. FIGURE 4.4.2 Graph for Problem 4647E48EIn Problems 110 write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L. 1. 9y 4y = sin xIn Problems 110 write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L. 2. y 5y = x2 2xIn Problems 110 write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L. 3. y 4y 12y = x 6In Problems 110 write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L. 4. 2y 3y 2y = 1In Problems 110 write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L. 5. y + 10y + 25y = ex6EIn Problems 110 write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L. 7. y + 2y 13y + 10y = xexIn Problems 110 write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L. 8. y + 4y + 3y = x2 cos x 3x9E10EIn Problems 1114 verify that the given differential operator annihilates the indicated functions. 11. D4; y = 10x3 2xIn Problems 1114 verify that the given differential operator annihilates the indicated functions. 12. 2D 1; y = 4ex/2In Problems 11-14 verify that the given differential operator annihilates the indicated functions. 13. (D 2)(D + 5); y = e2x + 3e5xIn Problems 11-14 verify that the given differential operator annihilates the indicated functions. 14. D2 + 64; y = 2 cos 8x 5 sin 8xIn Problems 15-26 find a linear differential operator that annihilates the given function. 15. 1 + 6x 2x3In Problems 15-26 find a linear differential operator that annihilates the given function. 16. x3(1 5x)In Problems 15-26 find a linear differential operator that annihilates the given function. 17. 1 + 7e2x18EIn Problems 15-26 find a linear differential operator that annihilates the given function. 19. cos 2xIn Problems 15-26 find a linear differential operator that annihilates the given function. 20. 1 + sin xIn Problems 15-26 find a linear differential operator that annihilates the given function. 21. 13x + 9x2 sin 4xIn Problems 1526 find a linear differential operator that annihilates the given function. 22. 8xsinx+10cos5xIn Problems 1526 find a linear differential operator that annihilates the given function. 23. ex+2xexx2exIn Problems 1526 find a linear differential operator that annihilates the given function. 24. (2ex)2In Problems 1526 find a linear differential operator that annihilates the given function. 25. 3+excos2xIn Problems 1526 find a linear differential operator that annihilates the given function. 26. exsinxe2xcosx27E28EIn Problems 27-34 find linearly independent functions that are annihilated by the given differential operator. 29. (D 6)(2D + 3)In Problems 27-34 find linearly independent functions that are annihilated by the given differential operator. 30. D2 9D 36In Problems 2734 find linearly independent functions that annihilated by the given differential operator. 31. D2+532EIn Problems 2734 find linearly independent functions that annihilated by the given differential operator. 33. D310D2+25DIn Problems 2734 find linearly independent functions that annihilated by the given differential operator. 34. D2(D5)(D7)In Problems 35-64 solve the given differential equation by undetermined coefficients. 35. y 9y = 54In Problems 3564 solve the given differential equation by undetermined coefficients. 36. 2y7y+5y=29In Problems 3564 solve the given differential equation by undetermined coefficients. 37. y+y=3In Problems 3564 solve the given differential equation by undetermined coefficients. 38. y+2y+y=10In Problems 3564 solve the given differential equation by undetermined coefficients. 39. y + 4y + 4y = 2x + 640EIn Problems 3564 solve the given differential equation by undetermined coefficients. 41. y + y = 8x2In Problems 3564 solve the given differential equation by undetermined coefficients. 42. y 2y + y = x3 +4xIn Problems 3564 solve the given differential equation by undetermined coefficients. 43. y y 12y = e4x44EIn Problems 35-64 solve the given differential equation by undetermined coefficients. 45. y 2y 3y = 4ex 9In Problems 35-64 solve the given differential equation by undetermined coefficients. 46. y + 6y + 8y = 3e2x + 2xIn Problems 35-64 solve the given differential equation by undetermined coefficients. 47. y + 25y = 6 sin xIn Problems 35-64 solve the given differential equation by undetermined coefficients. 48. y + 4y = 4 cos x + 3 sin x 8In Problems 35-64 solve the given differential equation by undetermined coefficients. 49. y + 6y + 9y = xe4xIn Problems 3564 solve the given differential equation by undetermined coefficients. 50. y + 3y 10y = x(ex + 1)In Problems 3564 solve the given differential equation by undetermined coefficients. 51. y y = x2ex + 5In Problems 35-64 solve the given differential equation by undetermined coefficients. 52. y + 2y + y = x2exIn Problems 35-64 solve the given differential equation by undetermined coefficients. 53. y 2y + 5y = ex sin xIn Problems 3564 solve the given differential equation by undetermined coefficients. 54. y+y+14y=ex(sin3xcos3x)In Problems 3564 solve the given differential equation by undetermined coefficients. 55. y + 25y = 20 sin 5xIn Problems 3564 solve the given differential equation by undetermined coefficients. 56. y + y = 4 cos x sin xIn Problems 3564 solve the given differential equation by undetermined coefficients. 57. y + y + y = x sin xIn Problems 3564 solve the given differential equation by undetermined coefficients. 58. y + 4y = cos2x59E60EIn Problems 3564 solve the given differential equation by undetermined coefficients. 61. y 3y + 3y y = ex x + 1662E63E64EIn Problems 6572 solve the given initial-value problem. 65. y 64y = 16, y(0) = 1, y(0) = 0In Problems 65-72 solve the given initial-value problem. 66. y + y = x, y(0) = 1, y(0) = 0In Problems 6572 solve the given initial-value problem. 67. y5y=x2,y(0)=0,y(0)=268EIn Problems 6572 solve the given initial-value problem. 69. y+y=8cos2x4sinx,y(/2)=1,y(/2)=0In Problems 6572 solve the given initial-value problem. 70. y 2y + y = xex + 5, y(0) = 2, y(0) = 2, y(0) = 171E72ESuppose L is a linear differential operator that factors but has variable coefficients. Do the factors of L commute? Defend your answer.In Problems 118 solve each differential equation by variation of parameters. 1. y+y=secxIn Problems 118 solve each differential equation by variation of parameters. 2. y+y=tanxIn Problems 118 solve each differential equation by variation of parameters. 3. y+y=sinxIn Problems 118 solve each differential equation by variation of parameters. 4. y+y=sectanIn Problems 1-18 solve each differential equation by variation of parameters. 5. y + y = cos2 xIn Problems 1-18 solve each differential equation by variation of parameters. 6. y + y = sec2 xIn Problems 1-18 solve each differential equation by variation of parameters. 7. y y = cosh xIn Problems 118 solve each differential equation by variation of parameters. 8. yy=sinh2xIn Problems 118 solve each differential equation by variation of parameters. 9. y9y=9xe3xIn Problems 118 solve each differential equation by variation of parameters. 10. 4yy=ex/2+3In Problems 1-18 solve each differential equation by variation of parameters. 11. y+3y+2y=11+exIn Problems 1-18 solve each differential equation by variation of parameters. 12. y2y+y=ex1+x2In Problems 1-18 solve each differential equation by variation of parameters. 13. y + 3y + 2y = sin exIn Problems 1-18 solve each differential equation by variation of parameters. 14. y 2y + y = et arctan tIn Problems 1-18 solve each differential equation by variation of parameters. 15. y + 2y + y = et ln tIn Problems 1-18 solve each differential equation by variation of parameters. 16. 2y + y = 6xIn Problems 1-18 solve each differential equation by variation of parameters. 17. 3y 6y + 6y = ex sec xIn Problems 1-18 solve each differential equation by variation of parameters. 18. 4y4y+y=ex/21x2In Problems 19-22 solve each differential equation by variation of parameters, subject to the initial conditions y(0) = 1, y(0) = 0. 19. 4y y = xex/2In Problems 19-22 solve each differential equation by variation of parameters, subject to the initial conditions y(0) = 1, y(0) = 0. 20. 2y +y y = x + 1In Problems 19-22 solve each differential equation by variation of parameters, subject to the initial conditions y(0) = 1, y(0) = 0. 21. y +2y 8y = 2e2x exIn Problems 19-22 solve each differential equation by variation of parameters, subject to the initial conditions y(0) = 1, y(0) = 0. 22. y 4y + 4y = (12x2 6x)e2xIn Problems 23-26 proceed as in Example 3 and solve each differential equation by variation of parameters. 23. y+y=ex224E25EIn Problems 23-26 proceed as in Example 3 and solve each differential equation by variation of parameters. 26. 2y + 2y + y = 4xIn Problems 27 and 28 the indicated functions are known linearly independent solutions of the associated homogeneous differential equation on (0, ). Find the general solution of the given nonhomogeneous equation. 27. x2y+xy+(x214)y=x3/2; y1= x1/2 cos x, y2 = x1/2 sin xIn Problems 27 and 28 the indicated functions are known linearly independent solutions of the associated homogeneous differential equation on (0, ). Find the general solution of the given nonhomogeneous equation. 28. x2y + xy + y = sec(ln x); y1 = cos(ln x), y2 = sin(ln x)In Problems 29-32 solve the given third-order differential equation by variation of parameters. 29. y + y = tan xIn Problems 2932 solve the given third-order differential equation by variation of parameters. 30. y + 4y = sec 2xIn Problems 29-32 solve the given third-order differential equation by variation of parameters. 31. y 2y + 2y = e4x32EIn Problems 33 and 34 discuss how the methods of undetermined coefficients and variation of parameters can be combined to solve thegiven differential equation. Carry out your ideas. 33. 3y" 6y' + 30y = 15 sin x + ex tan 3xIn Problems 33 and 34 discuss how the methods of undetermined coefficients and variation of parameters can be combined to solve thegiven differential equation. Carry out your ideas. 34. y 2y + y = 4x2 3 + x1ex35E36EIn Problems 118 solve the given differential equation. 1. x2y 2y = 0In Problems 118 solve the given differential equation. 2. 4x2y + y = 0In Problems 118 solve the given differential equation. 3. xy + y = 0In Problems 118 solve the given differential equation. 4. xy 3y = 0In Problems 118 solve the given differential equation. 5. x2y + xy + 4y = 0In Problems 118 solve the given differential equation. 6. x2y + 5xy + 3y = 0In Problems 118 solve the given differential equation. 7. x2y 3xy 2y = 0In Problems 118 solve the given differential equation. 8. x2y + 3xy 4y = 0In Problems 118 solve the given differential equation. 9. 25x2y + 25xy + y = 0