A.5.7: Section 5.7 Answers

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1. \(y_{p}=\frac{-\cos 3x\ln |\sec 3x+\tan 3x|}{9}\)

2. \(y_{p}=-\frac{\sin 2x\ln |\cos 2x|}{4}+\frac{x\cos 2x}{2}\)

3. \(y_{p} = 4e^{x} (1 + e^{x} ) \ln(1 + e^{−x} )\)

4. \(y_{p} = 3e^{x} (\cos x \ln | \cos x| + x \sin x)\)

5. \(y_{p}=\frac{8}{5}x^{7/2}e^{x}\)

6. \(y_{p}=e^{x}\ln (1-e^{-2x})-e^{-x}\ln (e^{2x}-1)\)

7. \(y_{p}=\frac{2(x^{2}-3)}{3}\)

8. \(y_{p}=\frac{e^{2x}}{x}\)

9. \(y_{p}=x^{1/2}e^{x}\ln x\)

10. \(y_{p}=e^{-x(x+2)}\)

11. \(y_{p}=-4x^{5/2}\)

12. \(y_{p} = −2x^{2} \sin x−2x \cos x\)

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

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

15. \(y_{p}=\frac{3x^{4}e^{x}}{2}\)

16. \(y_{p}=x^{a+1}\)

17. \(y_{p}=\frac{x^{2}\sin x}{2}\)

18. \(y_{p}=-2x^{2}\)

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

20. \(y_{p}=-\frac{\sqrt{x}}{2}\)

21. \(y_{p}=\frac{x^{3/2}}{4}\)

22. \(y_{p}=-3x^{2}\)

23. \(y_{p}=\frac{x^{3}e^{x}}{2}\)

24. \(y_{p}=-\frac{4x^{3/2}}{15}\)

25. \(y_{p}=x^{3}e^{x}\)

26. \(y_{p}=xe^{x}\)

27. \(y_{p}=x^{2}\)

28. \(y_{p} = xe^{x} (x − 2)\)

29. \(y_{p}=\sqrt{x}e^{x}(x-1)/4\)

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

38.

\(y=k_{0}\cosh x+k_{1}\sinh x+\int _{0}^{x}\sinh (x-t)f(t)dt\)
\(y'=k_{0}\sinh x+k_{1}\cosh x+\int _{0}^{x}\cosh (x-t)f(t)dt\)

39.

\(y(x)=k_{0}\cos x+k_{1}\sin x+\int _{0}^{x}\sin (x-t)f(t)dt\)
\(y'(x)=-k_{0}\sin x+k_{1}\cos x+\int_{0}^{x}\cos (x-t)f(t)dt\)