11.9: A.11.1- Section 11.1 Answers

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2. \(\lambda_{n}=n^{2},\quad y_{n}=\sin nx,\quad n=1,2,3,\ldots \)

3. \(\lambda_{0}=0,\quad y_{0}=1;\quad \lambda_{n}=n^{2},\quad y_{n}=\cos nx,\quad n=1,2,3,\ldots \)

4. \( \lambda_{n}=\frac{(2n-1)^{2}}{4},\quad y_{n}=\sin\frac{(2n-1)x}{2},\quad n=1,2,3,\ldots\)

5. \( \lambda_{n}=\frac{(2n-1)^{2}}{4},\quad y_{n}=\cos\frac{(2n-1)x}{2},\quad n=1,2,3,\ldots\)

6. \(\lambda_{0},\quad y_{0}=1,\quad \lambda_{n}=n^{2},\quad y_{1n}=\cos nx,\quad y_{2n}=\sin nx,\quad n=1,2,3,\ldots\)

7. \(\lambda_{n}=n^{2}\pi ^{2},\quad y_{n}=\cos n\pi x,\quad n=1,2,3,\ldots\)

8. \(\lambda_{n}=\frac{(2n-1)^{2}\pi ^{2}}{4},\quad y_{n}=\cos\frac{(2n-1)\pi x}{2},\quad n=1,2,3,\ldots\)

9. \(\lambda_{n}=n^{2}\pi ^{2},\quad y_{n}=\sin n\pi x,\quad n=1,2,3,\ldots\)

10. \(\lambda_{0}=0,\quad y_{0}=1,\quad \lambda_{n}=n^{2}\pi ^{2},\quad y_{1n}=\cos n\pi x,\quad y_{2n}=\sin n\pi xn\quad n=1,2,3,\ldots\)

11. \(\lambda_{n}=\frac{(2n-1)^{2}\pi ^{2}}{4},\quad y_{n}=\sin\frac{(2n-1)\pi x}{2},\quad n=1,2,3,\ldots\)

12. \(\lambda_{0},\quad y_{0}=1,\quad \lambda_{n}=\frac{n^{2}\pi ^{2}}{4},\quad y_{1n}=\cos\frac{n\pi x}{2},\quad y_{2n}=\sin\frac{n\pi x}{2},\quad n=1,2,3,\ldots\)

13. \(\lambda_{n}=\frac{n^{2}\pi ^{2}}{4},\quad y_{n}=\sin\frac{n\pi x}{2},\quad n=1,2,3,\ldots\)

14. \(\lambda_{n}=\frac{(2n-1)^{2}\pi ^{2}}{36},\quad y_{n}=\cos\frac{(2n-1)\pi x}{6},\quad n=1,2,3,\ldots\)

15. \(\lambda_{n}=(2n-1)^{2}\pi ^{2},\quad y_{n}=\sin (2n-1)\pi x,\quad n=1,2,3,\ldots\)

16. \(\lambda_{n}=\frac{n^{2}\pi ^{2}}{25},\quad y_{n}=\cos\frac{n\pi x}{5},\quad n=1,2,3,\ldots\)

23. \(\lambda_{n}=4n^{2}\pi ^{2}/L^{2}\quad y_{n}=\sin\frac{2n\pi x}{L},\quad n=1,2,3,\ldots\)

24. \(\lambda_{n}=n^{2}\pi ^{2}/L^{2}\quad y_{n}=\cos\frac{n\pi x}{L},\quad n=1,2,3,\ldots\)

25. \(\lambda_{n}=4n^{2}\pi ^{2}/L^{2}\quad y_{n}=\sin\frac{2n\pi x}{L},\quad n=1,2,3,\ldots\)

26. \(\lambda_{n}=n^{2}\pi ^{2}/L^{2}\quad y_{n}=\cos\frac{n\pi x}{L},\quad n=1,2,3,\ldots\)