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Mathematics LibreTexts

Section 5.1 Answers

  • Page ID
    28906
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    1. 

    (c) \(y=-2e^{2x}+e^{5x}\)

    (d) \(y=(5k_{0}-k_{1})\frac{e^{2x}}{3}+(k_{1}-2k_{0})\frac{e^{5x}}{3}\)

    2. 

    (c) \(y=e^{3x}(3\cos x-5\sin x)\)

    (d) \(y=e^{x}(k_{0}\cos x+(k_{1}-k_{0})\sin x\)

    3. 

    (c) \(y=e^{x}(7-3x)\)

    (d) \(y=e^{x}(k_{0}+(k_{1}-k_{0})x)\)

    4. 

    1. \(y=\frac{c_{1}}{x-1}+\frac{c_{2}}{x+1}\)
    2. \(y=\frac{2}{x-1}-\frac{3}{x+1};\: (-1,1)\)

    5. 

    1. \(e^{x}\)
    2. \(e^{2x}\cos x\)
    3. \(x^{2}+2x-2\)
    4. \(-\frac{5}{6}x^{-5/6}\)
    5. \(-\frac{1}{x^{2}}\)
    6. \((x\ln |x|)^{2}\)
    7. \(\frac{e^{2x}}{2\sqrt{x}}\)

    6. \(0\)

    7. \(W(x)=(1-x^{2})^{-1}\)

    8. \(W(x)=\frac{1}{x}\)

    10. \(y_{2}=e^{-x}\)

    11. \(y_{2}=xe^{3x}\)

    12. \(y_{2}=xe^{ax}\)

    13. \(y_{2}=\frac{1}{x}\)

    14. \(y_{2}=x\ln x\)

    15. \(y_{2}=x^{a}\ln x\)

    16. \(y_{2}=x^{1/2}e^{-2x}\)

    17. \(y_{2}=x\)

    18. \(y_{2}=x\sin x\)

    19. \(y_{2}=x^{1/2}\cos x\)

    20. \(y_{2}=xe^{-x}\)

    21. \(y_{2}=\frac{1}{x^{2}-4}\)

    22. \(y_{2}=e^{2x}\)

    23. \(y_{2}=x^{2}\)

    35. 

    1. \(y"-2xy'+5y=0\)
    2. \((2x-1)y"-4xy'+4y=0\)
    3. \(x^{2}y"-xy'+y=0\)
    4. \(x^{2}y"+xy'+y=0\)
    5. \(y"-y=0\)
    6. \(xy"-y'=0\)

    37. (c) \(y=k_{0}y_{1}+k_{1}y_{2}\)

    38. \(y_{1}=1,\: y_{2}=x-x_{0};\quad y=k_{0}+k_{1}(x-x_{0})\)

    39. \(y_{1}=\cosh (x-x_{0}),\:y_{2}=\sinh (x-x_{0});\: y=k_{0}\cosh (x-x_{0})+k_{1}\sinh (x-x_{0})\)

    40. \(y_{1}=\cos\omega (x-x_{0}),\: y_{2}=\frac{1}{\omega }\sin\omega (x-x_{0})y=k_{0}\cos\omega (x-x_{0})+\frac{k_{1}}{\omega }\sin\omega (x-x_{0})\)

    41. \(y_{1}=\frac{1}{1-x^{2}}\: y_{2}=\frac{x}{1-x^{2}}\: y=\frac{k_{0}+k_{1}x}{1-x^{2}}\)

    42.

    (c) \(k_{0}=k_{1}=0;\: y=\left\{\begin{array}{cc}{c_{1}x^{2}+c_{2}x^{3}}&{x\geq 0}\\{c_{1}x^{2}+c_{3}x^{3}}&{x<0}\end{array} \right.\)

    (d) \((0,\infty )\) if \(x_{0}>0,\) \((−∞, 0)\) if \(x_{0} < 0\)

    43. (c) \(k_{0}=0,\: k_{1}\) arbitrary \(y=k_{1}x+c_{2}x^{2}\)

    44.

    (c) \(k_{0}=k_{1}=0;\: y=\left\{\begin{array}{cc}{a_{1}x^{3}+a_{2}x^{4}}&{x\geq 0}\\{b_{1}x^{3}+b_{2}x^{4}}&{x<0}\end{array} \right.\)

    (d) \((0,\infty )\) if \(x_{0}>0,\) \((−∞, 0)\) if \(x_{0} < 0\)