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