

1.

(c) $$y=-2e^{2x}+e^{5x}$$

(d) $$y=(5k_{0}-k_{1})\frac{e^{2x}}{3}+(k_{1}-2k_{0})\frac{e^{5x}}{3}$$

2.

(c) $$y=e^{3x}(3\cos x-5\sin x)$$

(d) $$y=e^{x}(k_{0}\cos x+(k_{1}-k_{0})\sin x$$

3.

(c) $$y=e^{x}(7-3x)$$

(d) $$y=e^{x}(k_{0}+(k_{1}-k_{0})x)$$

4.

1. $$y=\frac{c_{1}}{x-1}+\frac{c_{2}}{x+1}$$
2. $$y=\frac{2}{x-1}-\frac{3}{x+1};\: (-1,1)$$

5.

1. $$e^{x}$$
2. $$e^{2x}\cos x$$
3. $$x^{2}+2x-2$$
4. $$-\frac{5}{6}x^{-5/6}$$
5. $$-\frac{1}{x^{2}}$$
6. $$(x\ln |x|)^{2}$$
7. $$\frac{e^{2x}}{2\sqrt{x}}$$

6. $$0$$

7. $$W(x)=(1-x^{2})^{-1}$$

8. $$W(x)=\frac{1}{x}$$

10. $$y_{2}=e^{-x}$$

11. $$y_{2}=xe^{3x}$$

12. $$y_{2}=xe^{ax}$$

13. $$y_{2}=\frac{1}{x}$$

14. $$y_{2}=x\ln x$$

15. $$y_{2}=x^{a}\ln x$$

16. $$y_{2}=x^{1/2}e^{-2x}$$

17. $$y_{2}=x$$

18. $$y_{2}=x\sin x$$

19. $$y_{2}=x^{1/2}\cos x$$

20. $$y_{2}=xe^{-x}$$

21. $$y_{2}=\frac{1}{x^{2}-4}$$

22. $$y_{2}=e^{2x}$$

23. $$y_{2}=x^{2}$$

35.

1. $$y"-2xy'+5y=0$$
2. $$(2x-1)y"-4xy'+4y=0$$
3. $$x^{2}y"-xy'+y=0$$
4. $$x^{2}y"+xy'+y=0$$
5. $$y"-y=0$$
6. $$xy"-y'=0$$

37. (c) $$y=k_{0}y_{1}+k_{1}y_{2}$$

38. $$y_{1}=1,\: y_{2}=x-x_{0};\quad y=k_{0}+k_{1}(x-x_{0})$$

39. $$y_{1}=\cosh (x-x_{0}),\:y_{2}=\sinh (x-x_{0});\: y=k_{0}\cosh (x-x_{0})+k_{1}\sinh (x-x_{0})$$

40. $$y_{1}=\cos\omega (x-x_{0}),\: y_{2}=\frac{1}{\omega }\sin\omega (x-x_{0})y=k_{0}\cos\omega (x-x_{0})+\frac{k_{1}}{\omega }\sin\omega (x-x_{0})$$

41. $$y_{1}=\frac{1}{1-x^{2}}\: y_{2}=\frac{x}{1-x^{2}}\: y=\frac{k_{0}+k_{1}x}{1-x^{2}}$$

42.

(c) $$k_{0}=k_{1}=0;\: y=\left\{\begin{array}{cc}{c_{1}x^{2}+c_{2}x^{3}}&{x\geq 0}\\{c_{1}x^{2}+c_{3}x^{3}}&{x<0}\end{array} \right.$$

(d) $$(0,\infty )$$ if $$x_{0}>0,$$ $$(−∞, 0)$$ if $$x_{0} < 0$$

43. (c) $$k_{0}=0,\: k_{1}$$ arbitrary $$y=k_{1}x+c_{2}x^{2}$$

44.

(c) $$k_{0}=k_{1}=0;\: y=\left\{\begin{array}{cc}{a_{1}x^{3}+a_{2}x^{4}}&{x\geq 0}\\{b_{1}x^{3}+b_{2}x^{4}}&{x<0}\end{array} \right.$$

(d) $$(0,\infty )$$ if $$x_{0}>0,$$ $$(−∞, 0)$$ if $$x_{0} < 0$$