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Mathematics LibreTexts

Section 9.2 Answers

  • Page ID
    29616
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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. \(y=c_{1}x^{r_{1}}+c_{2}x^{r_{2}}+c_{3}x^{r_{3}}\: (r_{1}, r_{2}, r_{3}\text{ distinct)};\: y=c_{1}x^{r_{1}}+(c_{2}+c_{3}\ln x)x^{r_{2}}\: (r_{1}, r_{2}\text{ distinct)};\: y=[c_{1}+c_{2}\ln x+c_{3}(\ln x)^{2}]x^{r_{1}};\: y=c_{1}x^{r_{1}}+x^{\lambda }[c_{2}\cos (\omega\ln x)+c_{3}\sin (\omega\ln x)]\)