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14.18: Section 5.1 Answers

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Aug 11, 2020
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- 14.17: Section 4.5 Answers
- 14.19: Section 5.2 Answers

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

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

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

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

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

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.

$y"-2xy'+5y=0$
$(2x-1)y"-4xy'+4y=0$
$x^{2}y"-xy'+y=0$
$x^{2}y"+xy'+y=0$
$y"-y=0$
$xy"-y'=0$

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) $k_{0}=k_{1}=0;\: y=\left\{\begin{array}{cc}{c_{1}x^{2}+c_{2}x^{3}}&{x\geq 0}\\[4pt]{c_{1}x^{2}+c_{3}x^{3}}&{x<0}\end{array} \right.$

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

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

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