Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Differential Equations with Boundary-Value Problems (MindTap Course List)
19RE20RE21RE22REIn 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)=0.7EUse 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)=112EIn 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 + 320EIn 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,(,)24EIn 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,)28E29EIn 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 31–34 verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval.
31.
32EIn 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 31–34 verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval.
33.
(a) Verify that and are, respectively, particular solutions of
and
(b) Use part (a) to find particular solutions of
and
36E37ESuppose that y1 = ex and y2 = ex are two solutions of a homogeneous linear differential equation. Explain why y3 = cosh x and y4 = sinh x are also solutions of the equation.(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.41E42EIn 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 x6EIn 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 = ex23E24EIn Problems 114 find the general solution of the given second-order differential equation. 1. 4y + y = 02E3E4E5E6EIn Problems 114 find the general solution of the given second-order differential equation. 7. 12y 5y 2y = 08EIn Problems 114 find the general solution of the given second-order differential equation. 9. y + 9y = 010E11E12E13EIn 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=020EIn Problems 1528 find the general solution of the given higher-order differential equation. 21. dPdt=p(1p);P=c1et1+c1et22EIn 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)=232E33E34E35E36E37E38E39EIn Problems 3740 solve the given boundary-value problem. 40. y2y+2y=0,y(0)=1,y()=1In Problems 41 and 42 solve the given problem first using the form of the general solution given in (10). Solve again, this time using the form given in (11). 41. y3y=0,y(0)=1,y(0)=5In Problems 41 and 42 solve the given problem first using the form of the general solution given in (10). Solve again, this time using the form given in (11). 42. yy=0,y(0)=1,y(1)=043E44E45E46E47E48EIn Problems 4958 find a homogeneous linear differential equation with constant coefficients whose general solution is given. 49. y=c1ex+c2e5x50E51E52E53E54E55E56E57E58ETwo 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?Find the general solution of 2 + 7y + 4y 4y 0 if m1=12 is one root of its auxiliary equation.Find the general solution of y + 6y + y 34y = 0 if it is known that y1 = e4x cos x is one solution.To solve y(4) + y = 0, we must find the roots of m4 + 1 = 0. This is a trivial problem using a CAS but can also be done by hand working with complex numbers. Observe that m4 + 1 = (m2 + 1)2 2m2. How does this help? Solve the differential equation.63E64EIn 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 2x9EIn Problems 1–26 solve the given differential equation by undetermined coefficients.
10. y″ + 2y′ = 2x + 5 − e−2x
In Problems 126 solve the given differential equation by undetermined coefficients. 11. yy+14y=3+ex/212EIn Problems 1–26 solve the given differential equation by undetermined coefficients.
13. y″ + 4y = 3 sin 2x
In 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 2x20EIn 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 + 2xex27E28E29EIn 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) = 132EIn 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) = 9236E37EIn Problems 3740 solve the given boundary-value problem. 38. y 2y + 2y = 2x 2, y(0) = 0, y() =39EIn 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 46In 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)=sin2x, (v) f(x)=exsinx, (vi) f(x)=sinx. 47. FIGURE 4.4.3 Graph for Problem 4748EIn 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 64EIn 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 = ex6E7E8EIn 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. 9. y(4) + 8y = 4In 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. 10. y(4) 8y + 16y = (x3 2x)e4xIn 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 2x316E17E18E19E20EIn 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+10cos5x23EIn 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. exsinxe2xcosxIn Problems 27-34 find linearly independent functions that are annihilated by the given differential operator. 27. D5In Problems 27-34 find linearly independent functions that are annihilated by the given differential operator. 28. D2 + 4DIn Problems 27-34 find linearly independent functions that are annihilated by the given differential operator. 29. (D 6)(2D + 3)30E31E32EIn Problems 2734 find linearly independent functions that annihilated by the given differential operator. 33. D310D2+25D34EIn 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=1039E40E41E42E43E44E45E46E47EIn Problems 35-64 solve the given differential equation by undetermined coefficients. 48. y + 4y = 4 cos x + 3 sin x 849E50EIn Problems 3564 solve the given differential equation by undetermined coefficients. 51. y y = x2ex + 552E53EIn 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 = cos2xIn Problems 3564 solve the given differential equation by undetermined coefficients. 59. y + 8y = 6x2 + 9x + 260E61E62E63E64E65E66E67E68E69EIn Problems 6572 solve the given initial-value problem. 70. y 2y + y = xex + 5, y(0) = 2, y(0) = 2, y(0) = 171E72E73EIn 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=sinx4EIn Problems 1-18 solve each differential equation by variation of parameters. 5. y + y = cos2 x6EIn 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+311E12EIn 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 t16EIn Problems 1-18 solve each differential equation by variation of parameters. 17. 3y 6y + 6y = ex sec x18EIn 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)e2x23EIn Problems 23-26 proceed as in Example 3 and solve each differential equation by variation of parameters. 24. y+4y=e2xxIn Problems 23-26 proceed as in Example 3 and solve each differential equation by variation of parameters. 25. y + y 2y = ln xIn 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)29EIn Problems 2932 solve the given third-order differential equation by variation of parameters. 30. y + 4y = sec 2x31E32EIn 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 3x34E35EFind the general solution of x4y+x3y4x2y=1 given that y1 = x2 is a solution of the associated homogeneous equation.In Problems 118 solve the given differential equation. 1. x2y 2y = 0In Problems 118 solve the given differential equation. 2. 4x2y + y = 03E4EIn Problems 118 solve the given differential equation. 5. x2y + xy + 4y = 06EIn Problems 118 solve the given differential equation. 7. x2y 3xy 2y = 08EIn Problems 118 solve the given differential equation. 9. 25x2y + 25xy + y = 0