Please solve the following problems with the method mentioned in their respective heading

Section 3.5: problem. 7, 12.

In Problems 1-26, solve the given differential equation by 28. 2y" + 3y' - 2y = 14x2 - 4x - 11, undetermined coefficients. y(0) = 0, y'(0) = 0 1. y" + 3y' + 2y = 6 29. 5y" + y' = -6x, y(0) = 0, y'(0) = -10 2 4y" + 9y = 15 30. y" + 4y' + 4y = (3 + x)e-2x, y(0) = 2, y'(0) = 5 3. y" - 10y' + 25y = 30x + 3 31. y" + 4y' + 5y = 35e-4x, y(0) = -3, y'(0) = 1 4. y" + y' - by = 2x 32. y" - y = cosh x, y(0) = 2, y'(0) = 12 5. Ay" ty'ty= x - 2x 6. y" - 8y' + 20y = 100x2 - 26xet 33 d x d:2 + w x = Fo sin wt, x(0) = 0, x' (0) = 0 7. y" + 3y = -48x23x 8. 4y" - 4y' - 3y = cos 2x 34 d'x 9. y" - y' = -3 d:2 + w x = Fo cos yt, x(0) = 0, x'(0) = 0 10. y" + 2y' = 2x + 5 - e-2x 35. y" - 2y" + y' = 2- 24e" + 40esx, y(0) = 2, y' (0) = 2, 11. y" - y' thy =3+ ex/2 12. y" - 16y = 24x y"(0) = -2 13. y" + 4y = 3 sin 2x 36. y" + 8y = 2x - 5 + 8e-2x, y(0) = -5, y' (0) = 3, y"(0) = -4 14. y" -4y = (x2 - 3) sin 2x 15. y" + y = 2x sin x In Problems 37-40, solve the given boundary-value problem. 16. y" - 5y' = 2x - 4x2 - x + 6 37. y" + y = x2 + 1, y(0) = 5, y(1) = 0 17. y" - 2y' + 5y = e" cos 2x 38. y" - 2y' + 2y = 2x - 2, y(0) = 0, y(TT) = TT 18. y" - 2y' + 2y = e"*(cos x - 3 sin x) 39. y" +3y = 6x, y(0) = 0, y(1) + y'(1) = 0 19. y" + 2y' + y = sin x + 3 cos 2x 40. y" + 3y = 6x, y(0) + y'(0) = 0, y(1) = 0 20. y" + 2y' -24y = 16 - (x + 2)ex 21. y" - 6y" = 3 - cos x In Problems 41 and 42, solve the given initial-value problem 22. y" - 2y" - 4y' + 8y = 6xe2x in which the input function g(x) is discontinuous. [Hint: Solve 23. y" - 3y" + 3y' - y = x - 4et each problem on two intervals, and then find a solution so that 24. y" - y" - "- 4y' + 4y = 5 - ex + e2x y and y' are continuous at x = 7/2 (Problem 41) and at x = 25. y(4) + 2y" + y = (x - 1)2 (Problem 42).] 26. y(4) - y" = 4x + 2xe-* 41. y" + 4y = g(x), y(0) = 1, y'(0) = 2, whereIn Problems 1-18, solve each differential equation by variation In Problems 19-22, solve each differential equation by of parameters. variation of parameters subject to the initial conditions 1. y" + y = sec x 2. y" + y = tan x y(0) = 1, y' (0) = 0. 3. y" + y = sin x 4. y" + y = sec 0 tan 0 19. 4y" - y = xet/ 5. y" + y = cos x 6. y" + y = sec x 20. 2y" +y' - y = x+ 1 7. y" - y = cosh x 8. y" - y = sinh 2x 21. y" + 2y' - 8y = 2e-2x - e-x 9x 9. y" - 4y = X 10. y" - 9y = 3x 22. y" - 4y' + 4y = (12x2 - 6x)e 2x 1 11. y" + 3y' + 2y = 1 + ex In Problems 23 and 24, the indicated functions are ex known linearly independent solutions of the associated 12. y" - 2y' + y =1+ x homogeneous differential equation on the interval (0, co). Find the general solution of the given nonhomogeneous 13. y" + 3y' + 2y = sine" 14. y" - 2y' + y = e' arctan t equation. 15. y" + 2y' + y = e"' Int 16. 2y" + 2y' + y = 4Vx 23. xy" + xy' + ( 12 - by = x322; y1 = x-1/2 cos x, 17. 3y" - 6y' + 6y = e' sec x 12 = x-1/2 -1/2 sin x 18. 4y" - 4y' + y = ex/2V1 - x2 24. xy' + xy' + y = sec(In x); y, = cos(In x), y2 = sin(In x)