# A.5.4: Section 5.4 Answers

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1. $$y_{p}=e^{3x}\left(-\frac{1}{4}+\frac{x}{2} \right)$$

2. $$y_{p}=e^{-3x}\left(1-\frac{x}{4}\right)$$

3. $$y_{p}=e^{x}\left(2-\frac{3x}{4}\right)$$

4. $$y_{p} = e^{2x} (1−3x+x^{2})$$

5. $$y_{p} = e^{−x} (1+x^{2} )$$

6. $$y_{p} = e^{x} (−2+x+ 2x^{2} )$$

7. $$y_{p}=xe^{-x}\left(\frac{1}{6}+\frac{x}{2} \right)$$

8. $$y_{p} = xe^{x} (1 + 2x)$$

9. $$y_{p}=xe^{3x}\left(-1+\frac{x}{2} \right)$$

10. $$y_{p} = xe^{2x} (−2+x)$$

11. $$y_{p}=x^{2}e^{-x}\left(1+\frac{x}{2} \right)$$

12. $$y_{p}=x^{2}e^{x}\left(\frac{1}{2}-x \right)$$

13. $$y_{p}=\frac{x^{2}e^{2x}}{2}(1-x+x^{2})$$

14. $$y_{p}=\frac{x^{2}e^{-x/3}}{27}(3-2x+x^{2})$$

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

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

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

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

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

20. $$y = −e^{2x} (1 + x) + 2e^{−x} − e^{5x}$$

21. $$y = xe^{2x} + 3e^{x} − e^{−4x}$$

22. $$y = e ^{-x} (2 + x − 2x^{2}) − e^{−3x}$$

23. $$y = e ^{-2x} (3 − x) − 2e^{5x}$$

24. $$y_{p}=-\frac{e^{x}}{3}(1-x)+e^{-x}(3+2x)$$

25. $$y_{p} = e^{x} (3 + 7x) + xe^{3x}$$

26. $$y_{p}= x^{3} e^{4x} + 1 + 2x + x^{2}$$

27. $$y_{p} = xe^{2x} (1 − 2x) + xe^{x}$$

28. $$y_{p} = e^{x} (1 + x) + x^{2} e^{−x}$$

29. $$y_{p} = x^{2} e^{−x} + e^{3x} (1 − x^{2} )$$

31. $$y_{p} = 2e^{2x}$$

32. $$y_{p}=5xe^{4x}$$

33. $$y_{p}=x^{2}e^{4x}$$

34. $$y_{p}=-\frac{e^{3x}}{4}(1+2x-2x^{2})$$

35. $$y_{p}=xe^{3x}(4-x+2x^{2})$$

36. $$y_{p} = x^{2} e^{−x/2} (−1 + 2x + 3x^{2} )$$

37.

1. $$y=e^{-x}\left(\frac{4}{3}x^{3/2}+c_{1}x+c_{2} \right)$$
2. $$y=e^{-3x}\left[\frac{x^{2}}{4}(2\ln x-3)+c_{1}x+c_{2} \right]$$
3. $$y=e ^{2x} [(x + 1) \ln |x + 1| + c_{1}x + c_{2}]$$
4. $$y=e^{-x/2}\left(x\ln |x| +\frac{x^{3}}{6}+c_{1}x+c_{2} \right)$$

39.

1. $$e^{x}(3+x)+c$$
2. $$-e^{-x}(1+x)^{2}+c$$
3. $$-\frac{e^{-2x}}{8}(3+6x+6x^{2}=4x^{3})+c$$
4. $$e^{x}(1 + x^{2} ) + c$$
5. $$e^{3x} (−6 + 4x + 9x^{2} ) + c$$
6. $$−e^{−x} (1 − 2x^{3} + 3x^{4} ) + c$$

40. $$\frac{(-1)^{k}k!e^{\alpha x}}{\alpha ^{k+1}}\sum_{r=0}^{k}\frac{(-\alpha x)^{r}}{r!}+c$$

This page titled A.5.4: Section 5.4 Answers is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.