A.9.2: Section 9.2 Answers

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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.

\(\{e^{x},xe^{x},e^{2x}\}\:\:1\)
\(\{\cos 2x, \sin 2x, e^{3x} \}\: :\: 26\)
\(\{e ^{−x} \cos x, e^{−x} \sin x, e^{x} \}\: :\: 5\)
\(\{1, x, x^{2}, e^{x} \}\: 2e^{x}\)
\(\{e^{x}, e^{−x}, \cos x, \sin x \}\:8\)
\(\{\cos x, \sin x, e^{x} \cos x, e^{x} \sin x\}\: :\: 5\)

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.

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

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