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1. $$y=e^{x}(c_{1}+c_{2}x+c_{3}x^{2})$$

2. $$y=c_{1}e^{x}+c_{2}e^{-x}+c_{3}\cos 3x+c_{4}\sin 3x$$

3. $$y = c_{1}e^{x} + c_{2} \cos 4x + c_{3} \sin 4x$$

4. $$y = c_{1}e^{x} + c_{2}e^{−x} + c_{3}e^{−3x/2}$$

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

6. $$y = c_{1}e^{x} + e^{x/2} (c_{2} + c_{3}x)$$

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

8. $$y = c_{1} + c_{2}x + c_{3} \cos x + c_{4} \sin x$$

9. $$y = c_{1}e^{2x} + c_{2}e^{−2x} + c_{3} \cos 2x + c_{4} \sin 2x$$

10. $$y = (c_{1} + c_{2}x) \cos \sqrt{6}x + (c_{3} + c_{4}x) \sin\sqrt{6}x$$

11. $$y = e^{3x/2} (c_{1} + c_{2}x) + e^{−3x/2} (c_{3} + c_{4}x)$$

12. $$y = c_{1}e^{−x/2} + c_{2}e^{−x/3} + c_{3} \cos x + c_{4} \sin x$$

13. $$y = c_{1}e^{x}+c_{2}e^{−2x}+c_{3}e^{−x/2}+c_{4}e^{−3x/2}$$

14. $$y = e^{x} (c_{1}+c_{2}x+c_{3} \cos x+c_{4} \sin x)$$

15. $$y = \cos 2x − 2 \sin 2x + e^{2x}$$

16. $$y = 2e^{x} + 3e^{−x} − 5e^{−3x}$$

17. $$y = 2e^{x} + 3xe^{x} − 4e^{−x}$$

18. $$y = 2e^{−x} \cos x − 3e^{−x} \sin x + 4e^{2x}$$

19. $$y = \frac{9}{5} e^{−5x/3} + e^{x} (1 + 2x)$$

20. $$y = e^{2x} (1 − 3x + 2x^{2} )$$

21. $$y = e^{3x} (2 − x) + 4e^{−x/2}$$

22. $$y = e^{x/2} (1 − 2x) + 3e^{−x/2}$$

23. $$y = \frac{1}{8} (5e^{2x} + e^{−2x} + 10 \cos 2x + 4 \sin 2x)$$

24. $$y = −4e^{x} + e^{2x} − e^{4x} + 2e^{−x}$$

25. $$y=2e^{x}-e^{-x}$$

26. $$y = e^{2x} + e^{−2x} + e^{−x} (3 \cos x + \sin x)$$

27. $$y = 2e^{−x/2} + \cos 2x − \sin 2x$$

28.

1. $$\{e^{x},xe^{x},e^{2x}\}\::\:1$$
2. $$\{\cos 2x, \sin 2x, e^{3x} \}\: :\: 26$$
3. $$\{e ^{−x} \cos x, e^{−x} \sin x, e^{x} \}\: :\: 5$$
4. $$\{1, x, x^{2} , e^{x} \}\: 2e^{x}$$
5. $$\{e^{x} , e^{−x} , \cos x, \sin x \}\:8$$
6. $$\{\cos x, \sin x, e^{x} \cos x, e^{x} \sin x\}\: :\: 5$$

29. $$\{e^{−3x} \cos 2x, e^{−3x} \sin 2x, e^{2x} , xe^{2x} , 1, x, x^{2} \}$$

30. $$\{e^{x} , xe^{x} , e^{x/2} , xe^{x/2} , x^{2} e^{x/2} , \cos x, \sin x \}$$

31. $$\{\cos 3x, x \cos 3x, x^{2} \cos 3x, \sin 3x, x \sin 3x, x^{2} \sin 3x, 1, x \}$$

32. $$\{e^{2x} , xe^{2x} , x^{2} e^{2x} , e^{−x} , xe^{−x} , 1 \}$$

33. $$\{\cos x, \sin x, \cos 3x, x \cos 3x, \sin 3x, x \sin 3x, e^{2x} \}$$

34. $$\{e^{2x} , xe^{2x} , e^{−2x} , xe^{−2x} , \cos 2x, x \cos 2x, \sin 2x, x \sin 2x \}$$

35. $$\{e^{−x/2} \cos 2x, xe^{−x/2} \cos 2x, x^{2} e^{−x/2} \cos 2x, e^{−x/2} \sin 2x, xe^{−x/2} \sin 2x, x^{2} e^{−x/2} \sin 2x \}$$

36. $$\{1, x, x^{2} , e^{2x} , xe^{2x} , \cos 2x, x \cos 2x, \sin 2x, x \sin 2x \}$$

37. $$\{\cos (x/2), x \cos (x/2), \sin (x/2), x \sin (x/2), \cos 2x/3 x \cos (2x/3), x^{2} \cos (2x/3), \sin (2x/3), x \sin (2x/3), x^{2} \sin (2x/3) \}$$

38. $$\{e^{−x} , e^{3x} , e^{x} \cos 2x, e^{x} \sin 2x \}$$

39. b. $$e^{(a_{1}+a_{2}+\ldots +a_{n})x}\prod_{1\leq i<j\leq n}(a_{j}-a_{i})$$

43.

1. $$\{ e^{x},e^{-x/2}\cos\left(\frac{\sqrt{3}}{2}x\right), e^{-x/2}\sin\left(\frac{\sqrt{3}}{2}x\right)\}$$
2. $$\{e^{-x},e^{x/2}\cos\left(\frac{\sqrt{3}}{2}x\right),e^{x/2}\sin\left(\frac{\sqrt{3}}{2}x\right)\}$$
3. $$\{e^{2x}\cos 2x,e^{2x}\sin 2x, e^{-2x}\cos 2x,e^{-2x}\sin 2x\}$$
4. $$\{e^{x},e^{-x},e^{x/2}\cos\left(\frac{\sqrt{3}}{2}x\right), e^{x/2}\sin\left(\frac{\sqrt{3}}{2}x\right), e^{-x/2}\cos\left(\frac{\sqrt{3}}{2}x\right), e^{-x/2}\sin\left(\frac{\sqrt{3}}{2}x\right)\}$$
5. $$\{\cos 2x,\sin 2x, e^{-\sqrt{3x}}\cos x, e^{-\sqrt{3x}}\sin x, e^{\sqrt{3x}}\cos x, e^{\sqrt{3x}}\sin x\}$$
6. $$\{1, e^{2x}, e^{3x/2}\cos\left(\frac{\sqrt{3}}{2}x\right), e^{3x/2}\sin\left(\frac{\sqrt{3}}{2}x\right), e^{x/2}\cos\left(\frac{\sqrt{3}}{2}x\right), e^{x/2}\sin\left(\frac{\sqrt{3}}{2}x\right)\}$$
7. $$\{e^{-x}. e^{x/2}\cos\left(\frac{\sqrt{3}}{2}x\right), e^{x/2}\sin\left(\frac{\sqrt{3}}{2}x\right), e^{-x/2}\cos\left(\frac{\sqrt{3}}{2}x \right), e^{-x/2}\sin\left(\frac{\sqrt{3}}{2}x\right)\}$$

45. $$y=c_{1}x^{r_{1}}+c_{2}x^{r_{2}}+c_{3}x^{r_{3}}\: (r_{1}, r_{2}, r_{3}\text{ distinct)};\: y=c_{1}x^{r_{1}}+(c_{2}+c_{3}\ln x)x^{r_{2}}\: (r_{1}, r_{2}\text{ distinct)};\: y=[c_{1}+c_{2}\ln x+c_{3}(\ln x)^{2}]x^{r_{1}};\: y=c_{1}x^{r_{1}}+x^{\lambda }[c_{2}\cos (\omega\ln x)+c_{3}\sin (\omega\ln x)]$$