11.33: A.5.2- Section 5.2 Answers

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