14.18: Section 5.1 Answers
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1.
(c) y=−2e2x+e5x
(d) y=(5k0−k1)e2x3+(k1−2k0)e5x3
2.
(c) y=e3x(3cosx−5sinx)
(d) y=ex(k0cosx+(k1−k0)sinx
3.
(c) y=ex(7−3x)
(d) y=ex(k0+(k1−k0)x)
4.
5.
6. 0
7. W(x)=(1−x2)−1
8. W(x)=1x
10. y2=e−x
11. y2=xe3x
12. y2=xeax
13. y2=1x
14. y2=xlnx
15. y2=xalnx
16. y2=x1/2e−2x
17. y2=x
18. y2=xsinx
19. y2=x1/2cosx
20. y2=xe−x
21. y2=1x2−4
22. y2=e2x
23. y2=x2
35.
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