A.5.5: Section 5.5 Answers

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1. \(y_{p}=\cos x+2\sin x\)

2. \(y_{p}=\cos x+(2-2x)\sin x\)

3. \(y_{p}=e^{x}(-2\cos x+3\sin x)\)

4. \(y_{p}=\frac{e^{2x}}{2}(\cos 2x-\sin 2x)\)

5. \(y_{p}=-e^{x}(x\cos x-\sin x)\)

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

7. \(y_{p} = x(\cos 2x − 3 \sin 2x)\)

8. \(y_{p} = −x [(2 − x) \cos x + (3 − 2x) \sin x]\)

9. \(y_{p}=x\left[x\cos\left(\frac{x}{2}\right)-3\sin\left(\frac{x}{2}\right) \right]\)

10. \(y_{p} = xe^{−x} (3 \cos x + 4 \sin x)\)

11. \(y_{p} = xe^{x} [(−1 + x) \cos 2x + (1 + x) \sin 2x]\)

12. \(y_{p} = −(14 − 10x) \cos x − (2 + 8x − 4x^{2} ) \sin x\)

13. \(y_{p} = (1 + 2x + x^{2}) \cos x + (1 + 3x^{2}) \sin x\)

14. \(y_{p}=\frac{x^{2}}{2}(\cos 2x-\sin 2x)\)

15. \(y_{p} = e^{x} (x^{2} \cos x + 2 \sin x)\)

16. \(y_{p} = e^{x} (1 − x^{2} )(\cos x + \sin x)\)

17. \(y_{p} = e^{x} (x^{2} − x^{3})(\cos x + \sin x)\)

18. \(y_{p} = e^{−x} [(1 + 2x) \cos x − (1 − 3x) \sin x]\)

19. \(y_{p} = x(2 \cos 3x − \sin 3x)\)

20. \(y_{p} = −x^{3} \cos x + (x + 2x^{2} ) \sin x\)

21. \(y_{p} = −e^{−x}[ (x + x^{2} ) \cos x − (1 + 2x) \sin x]\)

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

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

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

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

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

27. \(y_{p}=xe^{3x}-\frac{e^{x}}{5}(\cos x-2\sin x)\)

28. \(y_{p}=x(\cos x +2\sin x)-\frac{e^{x}}{2}(1-x)+\frac{e^{-x}}{2}\)

29. \(y_{p}=-\frac{xe^{x}}{2}(2+x)+2xe^{2x}+\frac{1}{10}(3\cos x+\sin x)\)

30. \(y_{p}=xe^{x}(\cos x+x\sin x)+\frac{e^{x}}{25}(4+5x)+1+x+\frac{x^{2}}{2}\)

31. \(y_{p}=\frac{x^{2}e^{2x}}{6}(3+x)-e^{2x}(\cos x-\sin x)+3e^{3x}+\frac{1}{4}(2+x)\)

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

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

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

40.

\(2x \cos x − (2 − x^{2}) \sin x + c\)
\(-\frac{e^{x}}{2}\left[ (1-x^{2})\cos x-(1-x)^{2}\sin x\right] +c\)
\(-\frac{e^{-x}}{25}[(4+10x)\cos 2x-(3-5x)\sin 2x]+c\)
\(-\frac{e^{-x}}{2}\left[ (1+x)^{2}\cos x-(1-x^{2})\sin x\right]+c\)
\(-\frac{e^{x}}{2}\left[x(3-3x+x^{2})\cos x-(3-3x+x^{3})\sin x \right]+c\)
\(-e^{x}[(1-2x)\cos x+(1+x)\sin x]+c\)
\(e^{-x}[x\cos x+x(1+x)\sin x]+c\)