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