A.5.6: Section 5.6 Answers

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1. \(y = 1 − 2x + c_{1}e^{−x} + c_{2}xe^{x} ;\: \{e^{−x}, xe^{x} \}\)

2. \(y=\frac{4}{3x^{2}}+c_{1}x+\frac{c_{2}}{x};\: \{x, 1/x\}\)

3. \(y=\frac{x(\ln |x|)^{2}}{2}+c_{1}x+c_{2}x\ln |x|;\: \{x, x\ln |x|\}\)

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

5. \(y=e^{x}\left(\frac{4}{5}x^{7/2}+c_{1}+c_{2}x \right);\: \{e^{x}, xe^{x}\}\)

6. \(y = e^{x} (2x^{3/2} + x^{1/2} \ln x + c_{1}x^{1/2} + c_{2}x^{−1/2} );\: \{x^{1/2} e^{x}, x^{−1/2}e^{−x}\}\)

7. \(y = e^{x} (x \sin x + \cos x \ln | \cos x| + c_{1} \cos x + c_{2} \sin x);\: \{e^{x} \cos x, e^{x} \sin x\}\)

8. \(y = e ^{−x ^{2}} (2e^{−2x} + c_{1} + c_{2}x);\: \{e^{ −x^{2}}, xe^{−x ^{2}} \}\)

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

10. \(y=\frac{xe^{2x}}{9}+xe^{-x}(c_{1}+c_{2}x);\: \{xe^{-x}, x^{2}e^{-x}\}\)

11. \(y=xe^{x}\left(\frac{x}{3}+c_{1}+\frac{c_{2}}{x^{2}} \right);\: \{xe^{x}, e^{x}/x\}\)

12. \(y=-\frac{(2x-1)^{2}e^{x}}{8}+c_{1}e^{x}+c_{2}xe^{-x};\: \{e^{x}, xe^{-x}\}\)

13. \(y=x^{4}+c_{1}x^{2}+c_{2}x^{2}\ln |x|; \{x^{2}, x^{2}\ln |x|\}\)

14. \(y = e^{−x} (x^{3/2} + c_{1} + c_{2}x ^{1/2} );\: \{e^{−x}, x^{1/2} e^{−x} \}\)

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

16. \(y=x^{1/2}\left(\frac{e^{2x}}{2}+c_{1}+c_{2}e^{x} \right);\:\{x^{1/2}, x^{1/2}e^{x}\}\)

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

18. \(\{e^{x}, e^{x}/x\}\)

19. \(\{x^{2}, x^{3}\}\)

20. \(\{\ln |x|, x\ln |x|\}\)

21. \(\{\sin\sqrt{x},\cos\sqrt{x}\}\)

22. \(\{e^{x}, x^{3}e^{x}\}\)

23. \(\{x^{a}, x^{a}\ln |x|\}\)

24. \(\{x\sin x,x\cos x\}\)

25. \(\{e^{2x}, x^{2}e^{2x}\}\)

26. \(\{x^{1/2}, x^{1/2}\cos x\}\)

27. \(\{x^{1/2}e^{2x}, x^{1/2}e^{-2x}\}\)

28. \(\{1/x, e^{2x}\}\)

29. \(\{e^{x}, x^{2}\}\)

30. \(\{e^{2x}, x^{2}e^{2x}\}\)

31. \(y=x^{4}=6x^{2}-8x^{2}\ln |x|\)

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

33. \(y=\frac{(x+1)}{4}[-e^{x}(3-2x)+7e^{-x}]\)

34. \(y=\frac{x^{2}}{4}+x\)

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

38.

\(y=\frac{-kc_{1}\sin kx+kc_{2}\cos kx}{c_{1}\cos kx+c_{2}\sin kx}\)
\(y=\frac{c_{1}+2c_{2}e^{x}}{c_{1}+c_{2}e^{x}}\)
\(y=\frac{-6c_{1}+c_{2}e^{7x}}{c_{1}+c_{2}e^{7x}}\)
\(y=-\frac{7c_{1}+c_{2}e^{6x}}{c_{1}+c_{2}e^{6x}}\)
\(y=-\frac{(7c_{1}-c_{2})\cos x+(c_{1}+7c_{2})\sin x}{c_{1}\cos x+c_{2}\sin x}\)
\(y=\frac{-2c_{1}+3c_{2}e^{5x/6}}{6(c_{1}+c_{2}e^{5x/6})}\)
\(y=\frac{c_{1}+c_{2}(x+6)}{6(c_{1}+c_{2}x)}\)

39.

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