9.4: Chapter 4 Answers

Section 4.1

1. c. \(y=-2e^{2x}+e^{5x}\)

2. c. \(y=e^{3x}(3\cos x-5\sin x)\)

3. c. \(y=e^{x}(7-3x)\)

4. a. \(y=\frac{c_{1}}{x-1}+\frac{c_{2}}{x+1}\)
b. \(y=\frac{2}{x-1}-\frac{3}{x+1};\: (-1,1)\)

5. \(e^{x}\)

6. \(e^{2x}\cos x\)

7. \(x^{2}+2x-2\)

8. \(-\frac{5}{6}x^{-5/6}\)

9. \(-\frac{1}{x^{2}}\)

10. \((x\ln |x|)^{2}\)

11. \(\frac{e^{2x}}{2\sqrt{x}}\)

23. a. \((0,2)\)

b. \((2,\pi)\)

24. a. \((\pi/2,3\pi/2)\)

b. \((1,\pi/2)\)

Section 4.2

1. \(y=-1+2x+3x^{2}+c_{1}e^{-6x}+c_{2}e^{x}\)

2. \(y=1+x+e^{2x}(c_{1}\cos x+c_{2}\sin x)\)

3. \(y=-x+x^{3}+c_{1}e^{-7x}+c_{2}e^{-x}\)

4. \(y = 1 − x^{2} + e^{2x} (c_{1} + c_{2}x)\)

5. \(y = 2x + x^{3} + e^{−x} (2 \cos 3x + 3 \sin 3x)\)

6. \(y = 1 + 2x + e^{−3x} (\cos x − \sin x)\)

7. \(y=c_{1}e^{-6x}+c_{2}e^{x}+\frac{e^{3x}}{3}\)

8. \(y =c_{1}e^{2x}\cos x + c_{2}e^{2x}\sin x+e^{2x}\)

9. \(y = −2e^{−2x} − e^{−7x} + e^{−x}\)

10. \(y = e^{x} + e^{2x} (1 − 3x)\)

11. \(y=\frac{4}{45}e^{x/2}+e^{-x}(c_{1}\cos 3x+c_{2}\sin 3x)\)

12. \(y = e^{−3x} (1 + c_{1}\cos x + c_{2}\sin x)\)

13. \(y = \cos x − \sin x + e^{4x} (c_{1} + c_{2}x)\)

14. \(y = \cos 2x − 2 \sin 2x + c_{1} + c_{2}e^{−x}\)

15. \(y=\cos 3x+e^{x}(c_{1}\cos\sqrt{2}x+c_{2}\sin\sqrt{2}x)\)

16. \(y = \cos x + \sin x + e^{−3x} (c_{1} \cos 2x + c_{2} \sin 2x)\)

17. \(y = −2 \cos 2x + \sin 2x + c_{1}e^{−4x} + c_{2}e^{−3x};\: y = −2 \cos 2x + \sin 2x + 2e^{−4x} − 3e^{−3x}\)

18. \(y = \cos 3x − \sin 3x + e^{3x} (c_{1} + c_{2}x)\: y = \cos 3x − \sin 3x + e^{3x} (1 + 2x)\)

19. \(y_{p}=-1+2x+3x^{2}+\frac{e^{3x}}{3};\: y=-1+2x+3x^{2}+\frac{e^{3x}}{3}+c_{1}e^{-6x}+c_{2}e^{x}\)

20. \(y_{p} = 1 + x + e^{2x};\: y = 1 + x + e^{2x} (1 + c_{1}\cos x + c_{2}\sin x)\)

21. \(y_{p} = −x + x^{3} − 2e^{−2x};\: y = −x + x^{3} − 2e^{−2x} + c_{1}e^{−7x} + c_{2}e^{−x}\)

22. \(y_{p} = 1 − x^{2} + e^{x};\: y = 1 − x^{2} + e^{x} + e^{2x} (c_{1} + c_{2}x)\)

23. \(y_{p}=2x+x^{3}+\frac{4}{45}e^{x/2};\: y=2x+x^{3}+\frac{4}{45}e^{x/2}+e^{-x}(c_{1}\cos 3x+c_{2}\sin 3x)\)

24. \(y_{p} = 1 +2x + e^{-3x};\: y = 1 +2x + e^{-3x} (1+c_{1}\cos x + c_{2}\sin x)\)

Section 4.3

1. \(y = c_{1}e^{−x} + c_{2}xe^{x}\)

2. \(y=c_{1}x+\frac{c_{2}}{x}\)

3. \(y=c_{1}x+c_{2}x\ln |x|\)

4. \(y =c_{1}e^{2x} + c_{2}e^{x}\)

5. \(y=c_{1}e^x+c_{2}xe^x\)

6. \(y =c_{1}x^{1/2}e^x + c_{2}x^{−1/2}e^x\)

7. \(y =c_{1} e^x \cos x + c_{2} e^x \sin x\)

8. \(y =c_{1}e^{-x^2} + c_{2}xe^{-x^2}\)

9. \(y =c_{1}x^{2} + \frac{c_{2}}{x^{2}}\)

10. \(y=c_{1}xe^{-x}+c_{2}x^2e^{-x}\)

11. \(y=c_{1}xe^{x}+{e^{x}\over x}\)

12. \(y=c_{1}e^{x}+c_{2}xe^{-x}\)

13. \(y=c_{1}x^{2}+c_{2}x^{2}\ln |x|\)

14. \(y =c_{1}e^{−x} + c_{2}x ^{1/2}e^{−x}\)

15. \(y =c_{1}e^{x}+c_{2}x^{2}e^{x}\)

16. \(y=c_{1}x^{1/2}+c_{2}x^{1/2}e^{x}\)

17. \(y =c_{1}x^{2} + c_{2}x^{4}\)

18. \(y=c_1e^x+c_2{e^{x}\over x}\)

19. \(y=c_1x^2+c_2 x^{3}\)

20. \(y=c_1\ln |x|+c_2x\ln |x|\)

21. \(y=c_1\sin\sqrt{x}+c_2\cos\sqrt{x}\)

22. \(y=c_1e^x+c_2x^{3}e^x\)

23. \(y=c_1x^a+c_2x^a\ln |x|\)

24. \(y=c_1x\sin x+c_2x\cos x\)

25. \(y=c_1e^{2x}+c_2x^2e^{2x}\)

26. \(y=c_1x^{1/2}+c_2x^{1/2}\cos x\)

27. \(y=c_1x^{1/2}e^{2x}+c_2x^{1/2}e^{-2x}\)

28. \(y=c_1{1\over x}+c_2e^{2x}\)

29. \(y=c_1e^x+c_2 x^2\)

30. \(y=c_1e^{2x}+c_2x^2e^{2x}\)

31. \(y=7x^{2}-6x^{2}\ln |x|\)

32. \(y={-1\over 2}(x+1)e^x+{3\over 2}(x+1)e^{-x}\)

33. \(y=-x^{-2}+x\)

Section 4.4

1. \(y=c_{1}e^{-6x}+c_{2}e^{x}\)

2. \(y=e^{2x}(c_{1}\cos x+c_{2}\sin x)\)

3. \(y=c_{1}e^{-7x}+c_{2}e^{-x}\)

4. \(y=e^{2x}(c_{1}+c_{2}x)\)

5. \(y=e^{-x}(c_{1}\cos 3x+c_{2}\sin 3x)\)

6. \(y=e^{-3x}(c_{1}\cos x+c_{2}\sin x)\)

7. \(y=e^{4x}(c_{1}+c_{2}x)\)

8. \(y=c_{1}+c_{2}e^{-x}\)

9. \(y=e^{x}(c_{1}\cos\sqrt{2}x+c_{2}\sin\sqrt{2}x)\)

10. \(y=e^{-3x}(c_{1}\cos 2x+c_{2}\sin 2x)\)

11. \(y=e^{-x/2}\left(c_{1}\cos\frac{3x}{2}+c_{2}\sin\frac{3x}{2} \right)\)

12. \(y=c_{1}e^{-x/5}+c_{2}e^{x/2}\)

13. \(y = e ^{-7x} (2 \cos x − 3 \sin x)\)

14. \(y=4e^{x/2}+6e^{-x/3}\)

15. \(y=3e^{x/3}-4e^{-x/2}\)

16. \(y=\frac{e^{-x/2}}{3}+\frac{3e^{3x/2}}{4}\)

17. \(y=e^{3x/2}(3-2x)\)

18. \(y=3e^{-4x}-4e^{-3x}\)

19. \(y=2xe^{3x}\)

20. \(y=e^{x/6}(3+2x)\)

21. \(y=e^{-2x}\left(3\cos\sqrt{6}x+\frac{2\sqrt{6}}{3}\sin\sqrt{6}x \right)\)

Section 4.5

1. \(y=e^{x}(c_{1}+c_{2}x+c_{3}x^{2})\)

2. \(y=c_{1}e^{x}+c_{2}e^{-x}+c_{3}\cos 3x+c_{4}\sin 3x\)

3. \(y = c_{1}e^{x} + c_{2} \cos 4x + c_{3} \sin 4x\)

4. \(y = c_{1}e^{x} + c_{2}e^{−x} + c_{3}e^{−3x/2}\)

5. \(y = c_{1}e^{−x} + e^{−2x} (c_{1} \cos x + c_{2} \sin x)\)

6. \(y = c_{1}e^{x} + e^{x/2} (c_{2} + c_{3}x)\)

7. \(y = e ^{-x/3} (c_{1} + c_{2}x + c_{3}x^{2} )\)

8. \(y = c_{1} + c_{2}x + c_{3} \cos x + c_{4} \sin x\)

9. \(y = c_{1}e^{2x} + c_{2}e^{−2x} + c_{3} \cos 2x + c_{4} \sin 2x\)

10. \(y = (c_{1} + c_{2}x) \cos \sqrt{6}x + (c_{3} + c_{4}x) \sin\sqrt{6}x\)

11. \(y = e^{3x/2} (c_{1} + c_{2}x) + e^{−3x/2} (c_{3} + c_{4}x)\)

12. \(y = c_{1}e^{−x/2} + c_{2}e^{−x/3} + c_{3} \cos x + c_{4} \sin x\)

13. \(y = c_{1}e^{x}+c_{2}e^{−2x}+c_{3}e^{−x/2}+c_{4}e^{−3x/2}\)

14. \(y = e^{x} (c_{1}+c_{2}x+c_{3} \cos x+c_{4} \sin x)\)

15. \(y = \cos 2x − 2 \sin 2x + e^{2x}\)

16. \(y = 2e^{x} + 3e^{−x} − 5e^{−3x}\)

17. \(y = 2e^{x} + 3xe^{x} − 4e^{−x}\)

18. \(y = 2e^{−x} \cos x − 3e^{−x} \sin x + 4e^{2x}\)

19. \(y = \frac{9}{5} e^{−5x/3} + e^{x} (1 + 2x)\)

20. \(y = e^{2x} (1 − 3x + 2x^{2} )\)

21. \(y = e^{3x} (2 − x) + 4e^{−x/2}\)

22. \(y = e^{x/2} (1 − 2x) + 3e^{−x/2}\)

23. \(y = \frac{1}{8} (5e^{2x} + e^{−2x} + 10 \cos 2x + 4 \sin 2x)\)

24. \(y = −4e^{x} + e^{2x} − e^{4x} + 2e^{−x}\)

25. \(y=2e^{x}-e^{-x}\)

26. \(y = e^{2x} + e^{−2x} + e^{−x} (3 \cos x + \sin x)\)

27. \(y = 2e^{−x/2} + \cos 2x − \sin 2x\)

Section 4.6

1. \(y=c_1e^{-3x}+c_2e^{3x}-6\)

2. \(y=c_1+c_2e^{-x}+3x\)

3. \(y=c_1e^{-2x}+c_2xe^{-2x}+{1\over 2}x+1\)

4. \(y=c_1e^{-3x}+c_2e^{4x}+{1\over 7}xe^{4x}\)

5. \(y=c_1e^{-x}+c_2e^{3x}-e^x+3\)

6. \(y=c_1\cos 5x+c_2\sin 5x+{1\over 4}\sin x\)

7. \(y=c_1e^{-3x}+c_2xe^{-3x}-{1\over 49}xe^{4x}+{2\over 343}e^{4x}\)

8. \(y=c_1e^{-x}+c_2e^{x}+{1\over 6}x^3e^x-{1\over 4}x^2e^x+{1\over 4}xe^x-5\)

9. \(y=c_1e^x\cos 2x+c_2e^x\sin 2x+{1\over 3}e^x\sin x\)

10. \(y=c_1\cos 5x+c_2\sin 5x-2x\cos 5x\)

11. \(y=c_1e^{-x/2}\cos {\sqrt 3\over 2}x+c_2e^{-x/2}\sin {\sqrt 3\over 2}x+\sin x+2\cos x-x\cos x\)

12. \(y=c_1e^x+c_2e^{2x}+e^{3x}\left(-\frac{1}{4}+\frac{x}{2} \right)\)

13. \(y=c_1e^x+c_2e^{5x}+e^{-3x}\left(1-\frac{x}{4}\right)\)

14. \(y=c_1e^{3x}+c_2e^{-x}+e^{x}\left(2-\frac{3x}{4}\right)\)

15. \(y=c_1e^{-x}+c_2xe^{-x}+e^{2x} (1−3x+x^{2})\)

16. \(y =c_1\cos 2x+c_2\sin 2x+e^{−x} (1+x^{2} )\)

17. \(y=c_1e^{2x}+c_2e^{-x}+e^{x} (−2+x+ 2x^{2} )\)

18. \(y=c_1e^{5x}+c_2e^{-x}+xe^{-x}\left(\frac{1}{6}+\frac{x}{2} \right)\)

19. \(y=c_1e^x+c_2e^{2x}+xe^{x} (1 + 2x)\)

20. \(y=c_1e^{-4x}+c_2e^{3x}+xe^{3x}\left(-1+\frac{x}{2} \right)\)

21. \(y= c_1e^{-x/2}+c_2e^{2x}+xe^{2x} (−2+x)\)

22. \(y= c_1e^{-x}+c_2xe^{-x}+x^{2}e^{-x}\left(1+\frac{x}{2} \right)\)

23. \(y= c_1e^x+c_2xe^{x}+x^{2}e^{x}\left(\frac{1}{2}-x \right)\)

24. \(y= c_1e^{2x}+c_2xe^{2x}+\frac{x^{2}e^{2x}}{2}(1-x+x^{2})\)

25. \(y= c_1e^{-x/3}+c_2xe^{-x/3}+\frac{x^{2}e^{-x/3}}{27}(3-2x+x^{2})\)

26. \(y=\frac{e^{3x}}{4}(-1+2x)+c_{1}e^{x}+c_{2}e^{2x}\)

27. \(y=e^{x}(1-2x)+c_{1}e^{2x}+c_{2}e^{4x}\)

28. \(y=\frac{e^{2x}}{5}(1-x)+e^{-3x}(c_{1}+c_{2}x)\)

29. \(y = xe^{x} (1 − 2x) + c_{1}e^{x} + c_{2}e^{−3x}\)

30. \(y = e^{x} \left[ x^{2} (1 − 2x) + c_{1} + c_{2}x\right ]\)

31. \(y=c_1e^{-x}+c_2e^{-2x}+\cos x+2\sin x\)

32. \(y=c_1e^{(-3+\sqrt 5)x/2}+c_2e^{(-3-\sqrt 5)x/2}+\cos x+(2-2x)\sin x\)

33. \(y=c_1e^{-x}+c_2xe^{-x}+e^{x}(-2\cos x+3\sin x)\)

34. \(y=c_1e^{(-3+\sqrt 17)x/2}+c_2e^{(-3-\sqrt 17)x/2}+\frac{e^{2x}}{2}(\cos 2x-\sin 2x)\)

35. \(y= c_1e^{x/2}\cos {\sqrt 3\over 2}x+c_2e^{x/2}\sin {\sqrt 3\over 2}x-e^{x}(x\cos x-\sin x)\)

36. \(y =c_1e^{(-3+\sqrt 17)x/2}+c_2e^{(-3-\sqrt 17)x/2}+ e^{−2x} (1 − 2x)(\cos 3x − \sin 3x)\)

37. \(y = c_1\cos 2x+c_2\sin 2x+x(\cos 2x − 3 \sin 2x)\)

38. \(y =c_1e^{x}+c_2xe^{x} + −x [(2 − x) \cos x + (3 − 2x) \sin x]\)

39. \(y=c_1\cos {x\over 2}+c_2\sin {x\over 2}+ x\left[x\cos\left(\frac{x}{2}\right)-3\sin\left(\frac{x}{2}\right) \right]\)

40. \(y = c_1e^{-x}\cos x+c_2e^{-x}\sin x+ xe^{−x} (3 \cos x + 4 \sin x)\)

41. \(y = c_1e^{x}\cos 2x+c_2e^{x}\sin 2x+xe^{x} [(−1 + x) \cos 2x + (1 + x) \sin 2x]\)

42. \(y =c_1e^{-x}+c_2xe^{-x} −(14 − 10x) \cos x − (2 + 8x − 4x^{2} ) \sin x\)

43. \(y = c_1e^{-x}+c_2e^{-2x} +(1 + 2x + x^{2}) \cos x + (1 + 3x^{2}) \sin x\)

44. \(y= c_1e^{-x}+c_2e^{-2x} +\frac{x^{2}}{2}(\cos 2x-\sin 2x)\)

45. \(y = c_1e^{2x}+c_2e^{3x} e^{x} (x^{2} \cos x + 2 \sin x)\)

46. \(y = c_1e^{x}+c_2xe^{x} +e^{x} (1 − x^{2} )(\cos x + \sin x)\)

47. \(y = c_1e^{x}\cos x+c_2e^{x}\sin x+e^{x} (x^{2} − x^{3})(\cos x + \sin x)\)

48. \(y =c_1e^{-x}+c_2xe^{-x} + e^{−x} [(1 + 2x) \cos x − (1 − 3x) \sin x]\)

49. \(y =c_1\cos 3x+c_2\sin 3x + x(2 \cos 3x − \sin 3x)\)

50. \(y= c_1e^{-x}+c_2e^{-2x}−x^{3} \cos x + (x + 2x^{2} ) \sin x\)

51. \(y = c_1e^{-x}+c_2e^{-3x}−e^{−x}[ (x + x^{2} ) \cos x − (1 + 2x) \sin x]\)

52. \(y={5\over 8}e^{-8x}+{5\over 8}e^{8x}-{1\over 4}\)

53. \(y=-{41\over 125}+{41\over 125}e^{5x}-{1\over 10}x^2+{9\over 25}x\)

54. \(y=-\pi\cos x-{11\over 3}\sin x-{8\over 3}\cos 2x+2x\cos x\)

55. \(y=2e^{2x}\cos 2x-{3\over 64}e^{2x}\sin 2x+{1\over 8}x^3+{3\over 16}x^2+{3\over 32}x\)

56. \(y = −e^{2x} (1 + x) + 2e^{−x} − e^{5x}\)

57. \(y = xe^{2x} + 3e^{x} − e^{−4x}\)

58. \(y = e ^{-x} (2 + x − 2x^{2}) − e^{−3x}\)

59. \(y = e ^{-2x} (3 − x) − 2e^{5x}\)

60. \(y = e^{x} (2 \cos x + 3 \sin x) + 3e^{x} − e^{6x}\)

61. \(y = e^{x} [(1 + 2x) \cos x + (1 − 3x) \sin x]\)

62. \(y = e^{x} (\cos x−2 \sin x)+e^{−3x} (\cos x+\sin x)\)

63. \(y = e^{3x} [(2 + 2x) \cos x − (1 + 3x) \sin x]\)

64. \(y = e^{3x} [(2 + 3x) \cos x + (4 − x) \sin x]+3e^{x}−5e^{2x}\)

65. \(y = xe^{−2x} \cos x + 3 \cos 2x\)

66. \(y=-\frac{3}{8}\cos 2x+\frac{1}{4}\sin 2x+e^{-x}-\frac{13}{8}e^{-2x}-\frac{3}{4}xe^{-2x}\)

Section 4.7

1. \(y=c_1\cos 3x+c_2\sin 3x+\frac{-\cos 3x\ln |\sec 3x+\tan 3x|}{9}\)

2. \(y= c_1\cos 2x+c_2\sin 2x-\frac{\sin 2x\ln |\cos 2x|}{4}+\frac{x\cos 2x}{2}\)

3. \(y = c_1e^x+c_2e^{2x}+4e^{x} (1 + e^{x} ) \ln(1 + e^{−x} )\)

4. \(y =c_1e^x\cos x+c_2e^x\sin x+3e^{x}+\cos x \ln | \cos x| + x \sin x\)

5. \(y=c_1e^x+c_2xe^x+\frac{8}{5}x^{7/2}e^{x}\)

6. \(y=c_1e^x+c_2e^{-x}+e^{x}\ln (1-e^{-2x})-e^{-x}\ln (e^{2x}-1)\)

7. \(y=c_1x+c_2{1\over x}+\frac{2(x^{2}-3)}{3}\)

8. \(y=c_1e^x+c_2{e^x\over x}+\frac{e^{2x}}{x}\)

9. \(y= c_1x^{1/2} e^x+c_2x^{-1/2}e^x +x^{1/2}e^{x}\ln x\)

10. \(y= c_1e^{-x^2}+c_2xe^{-x^2}+e^{-x(x+2)}\)

11. \(y= c_1x^2+c_2x^3 -4x^{5/2}\)

12. \(y=c_1x+c_2x^3 −2x^{2} \sin x−2x \cos x\)

13. \(y= c_1e^{-x}+c_2xe^x -\frac{xe^{-x}(x+1)}{2}\)

14. \(y= c_1\cos\sqrt x+c_2\sin\sqrt x -\frac{\sqrt{x}\cos\sqrt{x}}{2}\)

15. \(y= c_1e^x+c_2x^3e^x+ \frac{3x^{4}e^{x}}{2}\)

16. \(y= c_1x^a+c_2x^a \ln x +x^{a+1}\)

17. \(y=c_1 x\cos x+c_2x\sin x +\frac{x^{2}\sin x}{2}\)

18. \(y= c_1e^{x^2}+c_2e^{-x^2} -2x^{2}\)

19. \(y=c_1 e^{-x}+c_2e^{-x}\cos x -e^{-x}\sin x\)

20. \(y= c_1\sqrt xe^{2x}+c_2\sqrt xe^{-2x} -\frac{\sqrt{x}}{2}\)

21. \(y= c_1\sqrt x\sin x+c_2\sqrt x\cos x +\frac{x^{3/2}}{4}\)

22. \(y= c_1xe^x+c_2xe^{-x} -3x^{2}\)

23. \(y= c_1xe^x+c_2x^2e^x +\frac{x^{3}e^{x}}{2}\)

24. \(y= c_1{1\over x}++c_2x^3 -\frac{4x^{3/2}}{15}\)

25. \(y= c_1x^2+c_2x^2e^x +x^{3}e^{x}\)

26. \(y= c_1x^2e^x+c_2x^3e^x +xe^{x}\)

27. \(y= c_1x^2\cos x+c_2x^2\sin x +x^{2}\)

28. \(y =c_1 x+c_2e^x +xe^{x} (x − 2)\)

29. \(y= c_1\sqrt x+c_2\sqrt xe^x +\sqrt{x}e^{x}(x-1)/4\)

30. \(y=\frac{e^{2x}(3x^{2}-2x+6)}{6}+\frac{xe^{-x}}{3}\)

31. \(y = (x − 1)^{2} \ln(1 − x) + 2x^{2} − 5x + 3\)

32. \(y = (x^{2}−1)e^{x}−5(x−1)\)

33. \(y=\frac{x(x^{2}+6)}{3(x^{2}-1)}\)

34. \(y=-\frac{x^{2}}{2}+x+\frac{1}{2x^{2}}\)

35. \(y=\frac{x^{2}(4x+9)}{6(x+1)}\)

36. \(y=c_1e^x\cos 3x+c_2e^x\sin 3x+{2\over 17}\cos x+{9\over 17}\sin x-{1\over 27}e^x\cos 3x\ln |\sec 3x+\tan 3x|\)

37. \(y=c_1\cos x+c_2\sin x-{1\over 2}x\cos x+x\sin x+\cos x\ln |\cos x|\)

38. \(y=c_1e^{-x}+c_2e^{-2x}+3x^2-9x+{21\over 2}-e^{-2x}\sin e^x+(e^{-x}+e^{-2x})\ln (1+e^x)\)

39. \(y=c_1e^x+c_2xe^x+x^3+6x^2+22x+32-{1\over 2}e^x\ln(1+x^2)+xe^x\tan^{-1} x \)

40. \(y=c_1e^x+c_2xe^x+{1\over 2}x^2e^x+{1\over 2}x^2e^x\tan^{-1} x-{1\over 2}e^x\tan^{-1} x-{1\over 2}xe^x\ln (1+x^2)\)

Section 4.8 

1. \(y=x_{1}x^{-4}+c_{2}x^{-2}\)

2. \(y=c_{1}x+c_{2}x^{7}\)

3. \(y = x(c_{1} + c_{2} \ln x)\)

4. \(y = x^{−2} (c_{1} + c_{2} \ln x)\)

5. \(y = c_{1} \cos (\ln x) + c_{2} \sin (\ln x)\)

6. \(y = x^{2} [c_{1} \cos (3 \ln x) + c_{2} \sin (3 \ln x)]\)

7. \(y=c_{1}x+\frac{c_{2}}{x^{3}}\)

8. \(y = c_{1}x^{2/3} + c_{2}x^{3/4}\)

9. \(y = x^{−1/2} (c_{1} + c_{2} \ln x)\)

10. \(y = c_{1}x+c_{2}x^{1/3}\)

11. \(y = c_{1}x^{2}+c_{2}x^{1/2}\)

12. \(y =\frac{1}{x} [c_{1} \cos (2 \ln x) + c_{2} \sin (2 \ln x]\)

13. \(y = x^{−1/3} (c_{1} + c_{2} \ln x)\)

14. \(y = x [c_{1} \cos (3 \ln x) + c_{2} \sin (3 \ln x)]\)

15. \(y=c_{1}x^{3}+\frac{x_{2}}{x^{2}}\)

16. \(y=\frac{c_{1}}{x}+c_{2}x^{1/2}\)

17. \(y=x^{2}(c_{1}+c_{2}\ln x)\)

18. \(y=\frac{1}{x^{2}}\left[c_{1}\cos\left(\frac{1}{\sqrt{2}}\ln x \right)+c_{2}\sin\left(\frac{1}{\sqrt{2}}\ln x \right) \right]\)

19. \(y=c_1x^{-1}+c_2x^2\)

20. \(y=c_1+c_2\ln x\)

21. \(y=c_1\cos (2\ln x)+c_2\sin (2\ln x)\)

22. \(y=c_1x^{2-\sqrt 6}+c_2x^{2+\sqrt 6}\)

23. \(y=c_1\cos ({1\over 5}\ln x)+c_2\sin ({1\over 5}\ln x)\)

24. \(y=c_1x^{-2}+c_2x^{-2}\ln x\)

25. \(y=c_1x^{-1/2}\cos ({\sqrt 3\over 6}\ln x)+c_2x^{-1/2}\sin ({\sqrt 3\over 6}\ln x)\)

26. \(y=c_1x^{-3}+c_2x^1\)

27. \(y=c_1x^{(-3+\sqrt 5)/ 2}+c_2x^{(-3-\sqrt 5)/ 2}\)

28. \(y=c_1\cos({1\over 2}\ln x)+c_2\sin({1\over 2}\ln x)\)

29. \(y=c_1+c_2x^5+{1\over 5}x^5\ln x\)

30. \(y=c_1x+c_2x\ln x+x\ln^2 x\)

31. \(y=c_1x^{-1}+c_2x^3-{2\over 3}x^2\)

32. \(y=c_1x^{-1}+c_2x^3+{1\over 2}x^3\ln x\)

33. \(y=c_1x^2+c_2x^3+x^4-x^2\ln x\)

34. \(y=c_1x+c_2x^3+2x^2e^x-2xe^x\)

35. \(y=c_1x+c_2x\ln x+2+\ln x\)

36. \(y=c_1x^{-1/2}+c_2x^{-1}+{1\over 15}x^2+{1\over 6}x\)

37. \(y=c_1x^{-2}+c_2x^{-2}\ln x+{2\over 25}x^3\)

38. \(y=c_1x^{-1}+c_2x^{-8}+{1\over 30}x^2\)

39. \(y=c_1x^2+c_2x^3+{1\over 3}\ln x+{5\over 18}\)

40. \(y=c_1x^2\cos (3\ln x)+c_2x^2\sin (3\ln x)+{3\over 10}x+{4\over 13}\)

41. \(y=2-2x^{-2}\)

42. \(y=16x^2-2x^4\)

43. \(y=\cos (\ln x)+2\sin (\ln x)\)

44. \(y=5x^2-7x^2\ln x\)

45. \(y=x^{-1}\ln x\)

46. \(y={3\over 4}-\ln x+{1\over 4}x^2\)

47. \(y={1\over 16}x^2-{1\over 2}x^4+x^6\)

48. \(y=ex-ex^3+2x^2e^x-2xe^x\)

49. \(y=x\ln x+2+\ln x\)

Section 4.9

1. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=c_1e^{5t}-c_2e^{-t}\\y_2=2c_1e^{5t}+c_2e^{-t}\end{array}\)

 

2. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=-2c_1e^{t}+c_2e^{4t}\\y_2=c_1e^{t}+c_2e^{4t}\end{array}\)

 

3. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=2c_1e^{t}+2c_2e^{-3t}\\y_2=5c_1e^{t}+c_2e^{-3t}\end{array}\)

 

4. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=-2c_1e^{-7t/2}+4c_2e^{-t}\\y_2=c_1e^{-7t/2}+3c_2e^{-t}\end{array}\)

 

5. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=5c_1e^{8t}+c_2e^{-10t}\\y_2=2c_1e^{8t}+4c_2e^{-10t}\end{array}\)

 

6. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=c_1+2c_2e^{-5t}\\y_2=3c_1e^{5t}+c_2e^{-5t}\end{array}\)

 

7. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=c_1e^{3t}+c_2e^{-t}\\y_2=c_1e^{3t}-c_2e^{-t}\end{array}\)

 

8. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=c_1e^{-t/2}-c_2e^{-2t}\\y_2=c_1e^{-t/2}+c_2e^{-2t}\end{array}\)

 

9. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=-3c_1e^{-t}-c_2e^{-2t}\\y_2=c_1e^{-t}+2c_2e^{-2t}\end{array}\)

 

10. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=2c_1e^{-3t}-2c_2e^{t}\\y_2=c_1e^{-3t}+c_2e^{t}\end{array}\)

 

11. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=c_1e^{-2t}-4c_2e^{3t}\\y_2=c_1e^{-2t}+c_2e^{3t}\end{array}\)

 

12. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=3c_1e^{2t}+c_2e^{t}\\y_2=2c_1e^{2t}+c_2e^{t}\end{array}\)

 

13. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=-3c_1e^{-5t}-c_2e^{-3t}\\y_2=c_1e^{-5t}+c_2e^{-3t}\end{array}\)

 

14. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=c_1+3c_2e^{t}-11t-15\\y_2=c_1+2c_2e^{-}-11t-10\end{array}\)

 

15. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=c_1e^{t}+c_2e^{-t}+4t\\y_2=c_1e^{t}+3c_2e^{-t}+8t-4\end{array}\)

 

16. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=10c_1e^{3t/2}+2c_2e^{t/2}-\dfrac{13}{2}te^{t/2}-\dfrac{15}{2}e^{t/2}\\y_2=3c_1e^{3t/2}+c_2e^{t/2}-\dfrac{13}{4}te^{t/2}-\dfrac{9}{4}e^{t/2}\end{array}\)

 

17. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=-c_1e^{2t}\sin(2t)+c_2e^{2t}\cos(2t)-\dfrac{1}{8}e^{2t}\sin(2t)\cos(4t)-\dfrac{1}{8}e^{2t}\cos(2t)\cos(4t)\\y_2=2c_1e^{2t}\cos(2t)+2c_2e^{2t}\sin(2t)+\dfrac{1}{4}e^{2t}\cos(2t)\sin(4t)-\dfrac{1}{4}e^{2t}\sin(2t)\cos(4t)\end{array}\)

 

18. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=2c_1e^{t}+c_2e^{2t}+4te^{t}+3e^{t}\\y_2=c_1e^{t}+c_2e^{2t}+2te^{t}+3e^{t}\end{array}\)

 

19. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=2c_1e^{t}+c_2e^{2t}+\dfrac{1}{10}e^{-3t}-3\\y_2=c_1e^{t}+c_2e^{2t}-\dfrac{3}{20}e^{-3t}-1\end{array}\)

 

20. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=4c_1e^{3t}-2c_2e^{-3t}-12t-\dfrac{4}{3}\\y_2=c_1e^{3t}+c_2e^{-3t}-\dfrac{4}{3}\end{array}\)

 

21. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=4c_1e^{3t}-2c_2e^{-3t}-te^{t}-\dfrac{1}{4}e^{t}\\y_2=c_1e^{3t}+c_2e^{-3t}-\dfrac{1}{8}te^{-t}-\dfrac{1}{8}e^{t}\end{array}\)

 

22. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=-c_1e^{t}\sin(t)+c_2e^{t}\cos(t)-3e^{t}\\y_2=c_1e^{t}\cos(t)+c_2e^{t}\sin(t)+3e^{t}\end{array}\)

 

23. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=-c_1e^{t}\sin(t)+c_2e^{t}\cos(t)+te^{t}\cos(t)\\y_2=c_1e^{t}\cos(t)+c_2e^{t}\sin(t)+te^{t}\sin(t)\end{array}\)

 

24. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=3e^{t/2}\\y_2=2e^{-t/2}+3e^{t/2}\end{array}\)

 

25. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=-2e^{5t}+4e{-5t}\\y_2=-6e^{5t}+2e^{-5t}\end{array}\)

 

26. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=2e^{t/2}-2e^t\\y_2=-4e^{t/2}+e^{t}\end{array}\)

 

27. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=7e^{9t}-2e^{-3t}\\y_2=7e^{9t}-4e^{-3t}\end{array}\)

 

28. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=3e^{5t}-4e{-5t}\\y_2=9e^{5t}-2e^{-5t}\end{array}\)

 

29. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=2te^{2t}-e{2t}-2te^{4t}+2e^{4t}\\y_2=2te^{2t}+e{2t}+2te^{4t}\end{array}\)

 

30. \(\{y_1, y_2\}\quad \) where \(\quad \begin{array}\\ y_1=3t-1+\ln t\\y_2=3t-4+\ln t\end{array}\)