A.13.2: Section 13.2 Answers

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.

1. \(\lambda =0\) is an eigenvalue \(y_{0}=2-x\)
2. none
3. \(5.0476821,\: 14.9198790,\: 29.7249673,\: 49.4644528\quad y=2\sqrt{\lambda }\cos\sqrt{\lambda }x-\sin\sqrt{\lambda }x\)

12.

1. \(\lambda =0\) isn't an eigenvalue
2. \(−0.5955245\quad y=\cosh\sqrt{-\lambda }x\)
3. \(8.8511386,\: 38.4741053,\: 87.8245457,\: 156.9126094\quad y=\cos\sqrt{\lambda }x\)

13.

1. \(\lambda =0\) isn't an eigenvalue
2. none
3. \(0.1470328,\: 1.4852833,\: 4.5761411,\: 9.6059439\quad y=\sqrt{\lambda }\cos\sqrt{\lambda }x+\sin\sqrt{\lambda }x\)

14.

1. \(\lambda =0\) isn't an eigenvalue
2. \(−0.1945921\quad y=2\sqrt{-\lambda }\cosh\sqrt{-\lambda }x-\sinh\sqrt{-\lambda }x\)
3. \(1.9323619,\: 5.9318981,\: 11.9317920,\: 19.9317507\quad y=2\sqrt{\lambda }\cos\sqrt{\lambda }x-\sin\sqrt{\lambda }x\)

15.

1. \(\lambda =0\) isn't an eigenvalue
2. \(−1.0664054\quad y=\cosh\sqrt{-\lambda }x\)
3. \(1.5113188,\: 8.8785880,\: 21.2104662,\: 38.4805610\quad y=\cos\sqrt{\lambda }x\)

16.

1. \(\lambda =0\) isn't an eigenvalue
2. \(−1.0239346\quad y=\sqrt{-\lambda }\cosh\sqrt{-\lambda }x-\sinh\sqrt{-\lambda }x\)
3. \(2.0565705,\: 9.3927144,\: 21.7169130,\: 38.9842177\quad y=\sqrt{\lambda }\cos\sqrt{\lambda }x-\sin\sqrt{\lambda }x\)

17.

1. \(\lambda =0\) isn't an eigenvalue
2. \(−0.4357577\quad y=2\sqrt{-\lambda }\cosh\sqrt{-\lambda }x-\sinh\sqrt{-\lambda }x\)
3. \(0.3171423,\: 3.7055350,\: 9.1970150,\: 16.8760401\quad y=2\sqrt{\lambda }\cos\sqrt{\lambda }x-\sin\sqrt{\lambda }x\)

18.

1. \(\lambda =0\) isn't an eigenvalue
2. \(−2.1790546,\: −9.0006633\quad y=\sqrt{-\lambda }\cosh\sqrt{-\lambda }x-3\sinh\sqrt{-\lambda }x\)
3. \(5.8453181,\: 17.9260967,\: 35.1038567,\: 57.2659330\quad y=\sqrt{\lambda }\cos\sqrt{\lambda }x-3\sin\sqrt{\lambda }x\)

19.

1. \(\lambda =0\) is an eigenvalue \(y_{0}=2-x\)
2. \(−1.0273046\quad y=2\sqrt{-\lambda }\cosh\sqrt{-\lambda }x-\sinh\sqrt{-\lambda }x\)
3. \(8.8694608,\: 16.5459202,\: 26.4155505,\: 38.4784094\quad y=2\sqrt{\lambda }\cos\sqrt{\lambda }x-\sin\sqrt{\lambda }x\)

20.

1. \(\lambda =0\) isn't an eigenvalue
2. \(−7.9394171,\: −3.1542806\quad y=2\sqrt{-\lambda }\cosh\sqrt{-\lambda }x-5\sinh\sqrt{-\lambda }x\)
3. \(29.3617465,\: 78.777456,\: 147.8866417,\: 236.7229622\quad y=2\sqrt{\lambda }\cos\sqrt{\lambda }x-5\sin\sqrt{\lambda }x\)

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.

1. \(-L<\delta <0\)
2. \(\delta =-L\)

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.

1. \(y=x-\alpha \)
2. \(y=\alpha k\cosh kx-\sin kx\)
3. \(y=\alpha k\cos kx-\sin kx\)

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