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A.13.2: Section 13.2 Answers

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Jan 2, 2021
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- A.13.1: Section 13.1 Answers
- Index

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1. $(e^{bx}y')'+ce^{bx}y=0$

2. $(xy')'+\left(x-\frac{\nu ^{2}}{x}\right)y=0$

3. $(\sqrt{1-x^{2}}y')'+\frac{\alpha ^{2}}{\sqrt{1-x^{2}}}y=0$

4. $(x^{b}y')'+cx^{b-2}y=0$

5. $(e^{-x^{2}}y')'+2\alpha e^{-x^{2}}y=0$

6. $(xe^{-x}y')'+\alpha e^{-x}y=0$

7. $((1-x^{2})y')'+\alpha (\alpha +1)y=0$

9. $\lambda_{n}=n^{2}\pi ^{2},\quad y_{n}=e^{-x}\sin n\pi x\: (n=$ positive integer)

10. $\lambda_{0}=-1,\quad y_{0}=1\quad\lambda_{n}=n^{2}\pi ^{2},\quad y_{n}=e^{-x}(n\pi\cos n\pi x+\sin n\pi x)\: (n=$ positive integer)

11.

$\lambda =0$ is an eigenvalue $y_{0}=2-x$
none
$5.0476821,\: 14.9198790,\: 29.7249673,\: 49.4644528\quad y=2\sqrt{\lambda }\cos\sqrt{\lambda }x-\sin\sqrt{\lambda }x$

12.

$\lambda =0$ isn't an eigenvalue
$−0.5955245\quad y=\cosh\sqrt{-\lambda }x$
$8.8511386,\: 38.4741053,\: 87.8245457,\: 156.9126094\quad y=\cos\sqrt{\lambda }x$

13.

$\lambda =0$ isn't an eigenvalue
none
$0.1470328,\: 1.4852833,\: 4.5761411,\: 9.6059439\quad y=\sqrt{\lambda }\cos\sqrt{\lambda }x+\sin\sqrt{\lambda }x$

14.

$\lambda =0$ isn't an eigenvalue
$−0.1945921\quad y=2\sqrt{-\lambda }\cosh\sqrt{-\lambda }x-\sinh\sqrt{-\lambda }x$
$1.9323619,\: 5.9318981,\: 11.9317920,\: 19.9317507\quad y=2\sqrt{\lambda }\cos\sqrt{\lambda }x-\sin\sqrt{\lambda }x$

15.

$\lambda =0$ isn't an eigenvalue
$−1.0664054\quad y=\cosh\sqrt{-\lambda }x$
$1.5113188,\: 8.8785880,\: 21.2104662,\: 38.4805610\quad y=\cos\sqrt{\lambda }x$

16.

$\lambda =0$ isn't an eigenvalue
$−1.0239346\quad y=\sqrt{-\lambda }\cosh\sqrt{-\lambda }x-\sinh\sqrt{-\lambda }x$
$2.0565705,\: 9.3927144,\: 21.7169130,\: 38.9842177\quad y=\sqrt{\lambda }\cos\sqrt{\lambda }x-\sin\sqrt{\lambda }x$

17.

$\lambda =0$ isn't an eigenvalue
$−0.4357577\quad y=2\sqrt{-\lambda }\cosh\sqrt{-\lambda }x-\sinh\sqrt{-\lambda }x$
$0.3171423,\: 3.7055350,\: 9.1970150,\: 16.8760401\quad y=2\sqrt{\lambda }\cos\sqrt{\lambda }x-\sin\sqrt{\lambda }x$

18.

$\lambda =0$ isn't an eigenvalue
$−2.1790546,\: −9.0006633\quad y=\sqrt{-\lambda }\cosh\sqrt{-\lambda }x-3\sinh\sqrt{-\lambda }x$
$5.8453181,\: 17.9260967,\: 35.1038567,\: 57.2659330\quad y=\sqrt{\lambda }\cos\sqrt{\lambda }x-3\sin\sqrt{\lambda }x$

19.

$\lambda =0$ is an eigenvalue $y_{0}=2-x$
$−1.0273046\quad y=2\sqrt{-\lambda }\cosh\sqrt{-\lambda }x-\sinh\sqrt{-\lambda }x$
$8.8694608,\: 16.5459202,\: 26.4155505,\: 38.4784094\quad y=2\sqrt{\lambda }\cos\sqrt{\lambda }x-\sin\sqrt{\lambda }x$

20.

$\lambda =0$ isn't an eigenvalue
$−7.9394171,\: −3.1542806\quad y=2\sqrt{-\lambda }\cosh\sqrt{-\lambda }x-5\sinh\sqrt{-\lambda }x$
$29.3617465,\: 78.777456,\: 147.8866417,\: 236.7229622\quad y=2\sqrt{\lambda }\cos\sqrt{\lambda }x-5\sin\sqrt{\lambda }x$

21. $λ = 0,\quad y = xe^{−x}\quad 20.1907286,\: 118.8998692,\: 296.5544121,\: 553.1646458\quad y = e^{−x} \sin \sqrt{\lambda }x$

22. $λ_{n} = n^{2}\pi ^{2},\quad y_{n} = x \sin n\pi (x − 2)\quad (n =$ positive integer)

23. $λ = 0,\quad y = x(2 − x)\quad 20.1907286,\: 118.8998692,\: 296.5544121\: 553.1646458,\quad y = x \sin\sqrt{\lambda } (x − 2)$

24. $3.3730893,\: 23.1923372,\: 62.6797232,\: 121.8999231,\: 200.8578309\quad y = x \sin\sqrt{\lambda } (x − 1)$

25.

$-L<\delta <0$
$\delta =-L$

26. $\lambda _{0}=-1\alpha ^{2}\quad y_{0}=e^{-x/a}\quad\lambda _{n}=n^{2},\quad y_{n}=n\alpha\cos nx-\sin nx,\quad n=1,2,\ldots$

27.

$y=x-\alpha$
$y=\alpha k\cosh kx-\sin kx$
$y=\alpha k\cos kx-\sin kx$

29. b. $\lambda =-\alpha ^{2}/\beta ^{2}\quad y=e^{-\alpha x/\beta }$

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