A.5.1: Section 5.1 Answers

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 \(x_{0} < 0\)