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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.

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

39.

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