A.5.2: Section 5.2 Answers

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1. \(y=c_{1}e^{-6x}+c_{2}e^{x}\)

2. \(y=e^{2x}(c_{1}\cos x+c_{2}\sin x)\)

3. \(y=c_{1}e^{-7x}+c_{2}e^{-x}\)

4. \(y=e^{2x}(c_{1}+c_{2}x)\)

5. \(y=e^{-x}(c_{1}\cos 3x+c_{2}\sin 3x)\)

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

7. \(y=e^{4x}(c_{1}+c_{2}x)\)

8. \(y=c_{1}+c_{2}e^{-x}\)

9. \(y=e^{x}(c_{1}\cos\sqrt{2}x+c_{2}\sin\sqrt{2}x)\)

10. \(y=e^{-3x}(c_{1}\cos 2x+c_{2}\sin 2x)\)

11. \(y=e^{-x/2}\left(c_{1}\cos\frac{3x}{2}+c_{2}\sin\frac{3x}{2} \right)\)

12. \(y=c_{1}e^{-x/5}+c_{2}e^{x/2}\)

13. \(y = e ^{-7x} (2 \cos x − 3 \sin x)\)

14. \(y=4e^{x/2}+6e^{-x/3}\)

15. \(y=3e^{x/3}-4e^{-x/2}\)

16. \(y=\frac{e^{-x/2}}{3}+\frac{3e^{3x/2}}{4}\)

17. \(y=e^{3x/2}(3-2x)\)

18. \(y=3^{-4x}-4e^{-3x}\)

19. \(y=2xe^{3x}\)

20. \(y=e^{x/6}(3+2x)\)

21. \(y=e^{-2x}\left(3\cos\sqrt{6}x+\frac{2\sqrt{6}}{3}\sin\sqrt{6}x \right)\)

23. \(y=2e^{-(x-1)}-3e^{-2(x-1)}\)

24. \(y=\frac{1}{3}e^{-(x-2)}-\frac{2}{3}e^{7(x-2)}\)

25. \(y=e^{7(x-1)}(2-3(x-1))\)

26. \(y=e^{-(x-2)/3}(2-4(x-2))\)

27. \(y=2\cos\frac{2}{3}\left(x-\frac{\pi }{4} \right) -3\sin\frac{2}{3}\left(x-\frac{\pi }{4} \right)\)

28. \(y=2\cos\sqrt{3}\left( x-\frac{\pi }{3}\right)-\frac{1}{\sqrt{3}}\sin\sqrt{3}\left(x-\frac{\pi }{3} \right)\)

30. \(y=\frac{k_{0}}{r_{2}-r_{1}}\left(r_{2}e^{r_{1}(x-x_{0})}-r_{1}e^{r_{2}(x-x_{0})} \right) +\frac{k_{1}}{r_{2}-r_{1}}\left(e^{r_{2}(x-x_{0})}-e^{r_{1}(x-x_{0})} \right)\)

31. \(y=e^{r_{1}(x-x_{0})}[k_{0}+(k_{1}-r_{1}k_{0})(x-x_{0})]\)

32. \(y=e^{\lambda (x-x_{0})}\left[ k_{0}\cos\omega (x-x_{0})+\left(\frac{k_{1}-\lambda k_{0}}{\omega } \right)\sin\omega (x-x_{0}) \right]\)