# 11.34: A.5.3- Section 5.3 Answers

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

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

3. $$y_{p}=-x+x^{3};\: y=-x+x^{3}+c_{1}e^{-7x}+c_{2}e^{-x}$$

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

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

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

8. $$y_{p}=\frac{2}{x}$$

9. $$y_{p}=4x^{1/2}$$

10. $$y_{p}=\frac{x^{3}}{2}$$

11. $$y_{p}=\frac{1}{x^{3}}$$

12. $$y_{p}=9x^{1/3}$$

13. $$y_{p}=\frac{2x^{4}}{13}$$

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

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

18. $$y = −2e^{−2x};\: y = −2e^{−2x} + c_{1}e^{−7x} + c_{2}e^{−x};\: y = −2e^{−2x} − e^{−7x} + e^{−x}$$

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

20. $$y_{p}=\frac{4}{25}e^{x/2};\: y=\frac{4}{45}e^{x/2}+e^{-x}(c_{1}\cos 3x+c_{2}\sin 3x)$$

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

24. $$y_{p} = \cos x − \sin x;\: y = \cos x − \sin x + e^{4x} (c_{1} + c_{2}x)$$

25. $$y_{p} = \cos 2x − 2 \sin 2x;\: y = \cos 2x − 2 \sin 2x + c_{1} + c_{2}e^{−x}$$

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

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

28. $$y_{p} = −2 \cos 2x + \sin 2x;\: y = −2 \cos 2x + \sin 2x + c_{1}e^{−4x} + c_{2}e^{−3x};\: y = −2 \cos 2x + \sin 2x + 2e^{−4x} − 3e^{−3x}$$

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

30. $$y=\frac{1}{\omega _{0}^{2}-\omega ^{2}}(M\cos\omega x+N\sin\omega x)+c_{1}\cos\omega_{0}x+c_{2}\sin\omega_{0}x$$

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

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

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

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

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

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

This page titled 11.34: A.5.3- Section 5.3 Answers is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.