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Mathematics LibreTexts

A.9.2: Section 9.2 Answers

( \newcommand{\kernel}{\mathrm{null}\,}\)

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)]


This page titled A.9.2: Section 9.2 Answers is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.

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