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Mathematics LibreTexts

Section 12.3 Answers

  • Page ID
    30087
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    1. \(u(x,y)=\frac{8}{\pi ^{3}}\sum_{n=1}^{\infty}\frac{\sinh (2n-1)\pi (1-y)}{(2n-1)^{3}\sinh (2n-1)\pi }\sin (2n-1)\pi x\)

    2. \(u(x,y)=-\frac{32}{\pi ^{3}}\sum_{n=1}^{\infty}\frac{(1+(-1)^{n}2)\sinh n\pi (3-y)/2}{n^{3}\sinh 3n\pi /2}\sin \frac{n\pi x}{2}\)

    3. \(u(x,y)=\frac{8}{\pi ^{2}} \sum_{n=1}^{\infty}(-1)^{n+1}\frac{\sinh (2n-1)\pi (1-y/2)}{(2n-1)^{2}\sinh (2n-1)\pi}\sin\frac{(2n-1)\pi x}{2}\)

    4. \(u(x,y)=\frac{\pi }{2}\frac{\sinh (1-y)}{\sinh 1}\sin x-\frac{16}{\pi } \sum_{n=1}^{\infty}\frac{n\sinh 2n(1-y)}{(4n^{2}-1)^{2}\sinh 2n}\sin 2nx\)

    5. \(u(x,y)=3y+\frac{108}{\pi ^{3}} \sum_{n=1}^{\infty}(-1)^{n}\frac{\sinh n\pi y/3}{n^{3}\cosh 2n\pi /3}\cos\frac{n\pi x}{3}\)

    6. \(u(x,y)=\frac{y}{2}+\frac{4}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{\sinh (2n-1)\pi y}{(2n-1)^{3}\cosh 2(2n-1)\pi }\cos (2n-1)\pi x\)

    7. \(u(x,y)=-\frac{8y}{3}+\frac{32}{\pi ^{3}} \sum_{n=1}^{\infty}(-1)^{n}\frac{\sinh n\pi y/2}{n^{3}\cosh n\pi }\cos\frac{n\pi x}{2}\)

    8. \(u(x,y)=\frac{y}{3}+\frac{4}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{\sinh n\pi y}{n^{3}\cosh n\pi }\cos n\pi x\)

    9. \(u(x,y)=\frac{128}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{\cosh (2n-1)\pi (x-3)/4}{(2n-1)^{3}\cosh 3(2n-1)\pi /4}\sin\frac{(2n-1)\pi y}{4}\)

    10. \(u(x,y)=-\frac{96}{\pi ^{3}} \sum_{n=1}^{\infty}\left[ 1+(-1)^{n}\frac{4}{(2n-1)\pi }\right]\frac{\cosh (2n-1)\pi (x-2)/2}{(2n-1)^{3}\cosh (2n-1)\pi }\sin\frac{(2n-1)\pi y}{2}\)

    11. \(u(x,y)=\frac{768}{\pi ^{3}} \sum_{n=1}^{\infty}\left[ 1+(-1)^{n}\frac{2}{(2n-1)\pi }\right]\frac{\cosh (2n-1)\pi (x-2)/4}{(2n-1)^{3}\cosh (2n-1)\pi /2}\sin\frac{(2n-1)\pi y}{4}\)

    12. \(u(x,y)=\frac{96}{\pi ^{3}} \sum_{n=1}^{\infty}\left[ 3+(-1)^{n}\frac{4}{(2n-1)\pi }\right]\frac{\cosh 3(2n-1)\pi (x-2)/2}{(2n-1)^{3}\cosh (2n-1)\pi /2}\sin\frac{(2n-1)\pi y}{2}\)

    13. \(u(x,y)=-\frac{16}{\pi }\sum_{n=1}^{\infty}\frac{\cosh (2n-1)x/2}{(2n-3)(2n+1)(2n-1)\sinh (2n-1)/2}\cos \frac{(2n-1)y}{2}\)

    14. \(u(x,y)=-\frac{432}{\pi ^{3}}\sum_{n=1}^{\infty}\left[ 1+\frac{4(-1)^{n}}{(2n-1)\pi }\right] \frac{\cosh (2n-1)\pi x/6}{(2n-1)^{3}\sinh (2n-1)\pi /3}\cos\frac{(2n-1)\pi y}{6}\)

    15. \(u(x,y)=-\frac{64}{\pi }\sum_{n=1}^{\infty}(-1)^{n}\frac{\cosh (2n-1)x/2}{(2n-1)^{4}\sinh (2n-1)/2}\cos\frac{(2n-1)y}{2}\)

    16. \(u(x,y)=-\frac{192}{\pi ^{4}}\sum_{n=1}^{\infty}\frac{\cosh (2n-1)\pi x/2}{(2n-1)^{4}\sinh (2n-1)\pi /2}\left[(-1)^{n}+\frac{2}{(2n-1)\pi }\right]\cos\frac{(2n-1)\pi y}{2}\)

    17. \(u(x,y)= \sum_{n=1}^{\infty}\alpha_{n}\frac{\sinh n\pi y/a}{\sinh n\pi b/a}\sin\frac{n\pi x}{a},\quad \alpha_{n}=\frac{2}{a}\int_{0}^{a}f(x)\sin\frac{n\pi x}{a}dx,\quad u(x,y)=\frac{72}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{\sinh (2n-1)\pi y/3}{(2n-1)^{3}\sinh 2(2n-1)\pi /3}\sin\frac{(2n-1)\pi x}{3}\)

    18. \(u(x,y)=\alpha_{0}(1-y/b)+ \sum_{n=1}^{\infty}\alpha_{n}\frac{\sinh n\pi (b-y)/a}{\sinh n\pi b/a}\cos\frac{n\pi x}{a},\quad \alpha_{0}=\frac{1}{a}\int_{0}^{a}f(x)dx,\quad \alpha_{n}=\frac{2}{a}\int_{0}^{a}f(x)\cos\frac{n\pi x}{a}dx,\quad n\geq 1,\quad u(x,y)=\frac{8(1-y)}{15}-\frac{48}{\pi ^{4}} \sum_{n=1}^{\infty}\frac{\sinh n\pi (1-y)}{\sinh n\pi }\cos n\pi x\)

    19. \(u(x,y)= \sum_{n=1}^{\infty}\alpha_{n}\frac{\sinh (2n-1)\pi (b-y)/2a}{\sinh (2n-1)\pi b/a}\cos\frac{(2n-1)\pi x}{2a},\quad \alpha_{n}=\frac{2}{a}\int_{0}^{a}f(x)\cos\frac{(2n-1)\pi x}{2a}dx,\quad u(x,y)=\frac{288}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{\sinh (2n-1)\pi (2-y)/6}{(2n-1)^{3}\sinh (2n-1)\pi /3}\sin\frac{(2n-1)\pi x}{6}\)

    20. \(u(x,y)= \sum_{n=1}^{\infty}\alpha_{n}\frac{\sinh (2n-1)\pi (b-y)/2a}{\sinh (2n-1)\pi b/2a}\sin\frac{(2n-1)\pi x}{2a},\quad \alpha_{n}=\frac{2}{a}\int_{0}^{a}f(x)\sin\frac{(2n-1)\pi x}{2a}dx,\quad u(x,y)=\frac{32}{\pi ^{3}} \sum_{n=1}^{\infty}\left[ (-1)^{n}5+\frac{18}{(2n-1)\pi }\right]\frac{\sinh (2n-1)\pi (2-y)/2}{(2n-1)^{3}\sinh (2n-1)\pi }\cos\frac{(2n-1)\pi x}{2}\)

    21. \(u(x,y)= \sum_{n=1}^{\infty}\alpha_{n}\frac{\cosh n\pi (y-b)/a}{\cosh n\pi b/a}\sin\frac{n\pi x}{a},\quad \alpha_{n}=\frac{2}{a}\int_{0}^{a}f(x)\sin\frac{n\pi x}{a}dx,\quad u(x,y)=-12 \sum_{n=1}^{\infty}(-1)^{n}\frac{\cosh n(y-2)}{n^{3}\cosh 2n}\sin nx\)

    22. \(u(x,y)=\alpha_{0}+ \sum_{n=1}^{\infty}\alpha_{n}\frac{\cosh n\pi y/a}{\cosh n\pi b/a}\cos\frac{n\pi x}{a},\quad \alpha_{0}=\frac{1}{a}\int_{0}^{a}f(x)dx,\quad \alpha_{n}=\frac{2}{a}\int_{0}^{a}f(x)\cos\frac{n\pi x}{a}dx,\quad n\geq 1,\quad u(x,y)=\frac{\pi ^{4}}{30}-3 \sum_{n=1}^{\infty}\frac{1}{n^{4}}\frac{\cosh 2ny}{\cos 2n}\cos 2nx\)

    23. \(u(x,y)=\frac{a}{\pi } \sum_{n=1}^{\infty}\alpha_{n}\frac{\sinh n\pi (y-b)/a}{n\cosh n\pi b/a}\sin\frac{n\pi x}{a},\quad \alpha_{n}=\frac{2}{a}\int_{0}^{a}f(x)\sin\frac{n\pi x}{a}dx,\quad u(x,y)=\frac{4}{\pi } \sum_{n=1}^{\infty}(-1)^{n+1}\frac{\sinh (2n-1)(y-1)}{(2n-1)^{3}\cosh (2n-1)}\sin (2n-1)x\)

    24. \(u(x,y)= \sum_{n=1}^{\infty}\alpha _{n}\frac{\cosh n\pi x/b}{\cosh n\pi a/b}\sin\frac{n\pi y}{b},\quad\alpha_{n}=\frac{2}{b}\int_{0}^{b}g(y)\sin\frac{n\pi y}{b}dy,\quad u(x,y)=\frac{96}{\pi ^{5}} \sum_{n=1}^{\infty}\frac{\cosh (2n-1)\pi x}{(2n-1)^{5}\cosh (2n-1)\pi }\sin (2n-1)\pi y\)

    25. \(u(x,y)= \sum_{n=1}^{\infty}\alpha_{n}\frac{\cosh (2n-1)\pi x/2b}{\cosh (2n-1)\pi a/2b}\cos\frac{(2n-1)\pi y}{2b},\quad \alpha_{n}=\frac{2}{b}\int_{0}^{b}g(y)\cos\frac{(2n-1)\pi y}{2b}dy,\quad u(x,y)=-\frac{128}{\pi ^{3}} \sum_{n=1}^{\infty}(-1)^{n}\frac{\cosh (2n-1)\pi x/4}{(2n-1)^{3}\cosh (2n-1)\pi /2}\cos\frac{(2n-1)\pi y}{4}\)

    26. \(u(x,y)=\frac{b}{\pi } \sum_{n=1}^{\infty}\alpha_{n}\frac{\cosh n\pi x/b}{n\sinh n\pi a/b}\sin\frac{n\pi y}{b},\quad\alpha_{n}=\frac{2}{b}\int_{0}^{b}g(y)\sin\frac{n\pi y}{b}dy,\quad u(x,y)=\frac{64}{\pi ^{3}} \sum_{n=1}^{\infty}(-1)^{n+1}\frac{\cosh (2n-1)\pi x/4}{(2n-1)^{3}\sinh (2n-1)\pi /4}\sin\frac{(2n-1)\pi y}{4}\)

    27. \(u(x,y)=-\frac{2b}{\pi } \sum_{n=1}^{\infty}\alpha_{n}\frac{\cosh (2n-1)\pi (x-a)/2b}{(2n-1)\sinh (2n-1)\pi a/2b}\sin\frac{(2n-1)y}{2b},\quad \alpha_{n}=\frac{2}{b}\int_{0}^{b}g(y)\sin\frac{(2n-1)\pi y}{2b}dy,\quad u(x,y)=192 \sum_{n=1}^{\infty}\left[1+(-1)^{n}\frac{4}{(2n-1)\pi}\right]\frac{\cosh (2n-1)(x-1)/2}{(2n-1)^{4}\sinh (2n-1)/2}\sin\frac{(2n-1)y}{2}\)

    28. \(u(x,y)=\alpha_{0}(x-a)+\frac{b}{\pi } \sum_{n=1}^{\infty}\alpha_{n}\frac{\sinh n\pi (x-a)/b}{n\cosh n\pi a/b}\cos\frac{n\pi y}{b},\quad\alpha_{0}=\frac{1}{b}\int_{0}^{b}g(y)\cos\frac{n\pi y}{b}dy,\quad\alpha_{n}=\frac{2}{b}\int_{0}^{b}g(y)\cos\frac{n\pi y}{b}dy,\quad u(x,y)=\frac{\pi (x-2)}{2}-\frac{4}{\pi } \sum_{n=1}^{\infty}\frac{\sinh (2n-1)(x-2)}{(2n-1)^{3}\cosh 2(2n-1)}\cos (2n-1)y\)

    29. \(u(x,y)=\alpha_{0}+ \sum_{n=1}^{\infty}\alpha_{n}e^{-n\pi y/a}\cos\frac{n\pi x}{a},\quad \alpha_{0}=\frac{1}{a}\int_{0}^{a}f(x)dx,\quad \alpha_{n}=\frac{2}{a}\int_{0}^{a}f(x)\cos\frac{n\pi x}{a}dx,\quad n\geq 1,\quad u(x,y)=\frac{\pi ^{3}}{2}-\frac{48}{\pi } \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}e^{-(2n-1)y}\cos (2n-1)x\)

    30. \(u(x,y)= \sum_{n=1}^{\infty}\alpha_{n}e^{-(2n-1)\pi y/2a}\cos\frac{(2n-1)\pi x}{2a},\quad\alpha_{n}=\frac{2}{a}\int_{0}^{a}f(x)\cos\frac{(2n-1)\pi x}{2a}dx,\quad u(x,y)=-\frac{288}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{(2n-1)^{3}}e^{-(2n-1)\pi y/6}\cos\frac{(2n-1)\pi x}{6}\)

    31. \(u(x,y)= \sum_{n=1}^{\infty}\alpha_{n}e^{-(2n-1)\pi y/2a}\sin\frac{(2n-1)\pi x}{2a},\quad\alpha_{n}=\frac{2}{a}\int_{0}^{a}f(x)\sin\frac{(2n-1)\pi x}{2a}dx,\quad u(x,y)=\frac{32}{\pi } \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}e^{-(2n-1)y/2}\sin\frac{(2n-1)x}{2}\)

    32. \(u(x,y)=-\frac{a}{\pi } \sum_{n=1}^{\infty}\frac{\alpha_{n}}{n}e^{-n\pi y/a}\sin\frac{n\pi x}{a},\quad\alpha_{n}=\frac{2}{a}\int_{0}^{a}f(x)\sin\frac{n\pi x}{a}dx,\quad u(x)=4 \sum_{n=1}^{\infty}\frac{(1+(-1)^{n}2)}{n^{4}}e^{-ny}\sin nx\)

    33. \(u(x,y)=-\frac{2a}{\pi } \sum_{n=1}^{\infty}\frac{\alpha_{n}}{2n-1}e^{-(2n-1)\pi y/2a}\cos\frac{(2n-1)\pi x}{2a},\quad\alpha_{n}=\frac{2}{a}\int_{0}^{a}f(x)\cos\frac{(2n-1)\pi x}{2a}dx,\quad u(x,y)=\frac{5488}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\left[ 1+\frac{4(-1)^{n}}{(2n-1)\pi }\right]e^{-(2n-1)\pi y/14}\cos\frac{(2n-1)\pi x}{14}\)

    34. \(u(x,y)=-\frac{2a}{\pi } \sum_{n=1}^{\infty}\frac{\alpha _{n}}{2n-1}e^{-(2n-1)\pi y/2a}\sin\frac{(2n-1)\pi x}{2a},\quad\alpha_{n}=\frac{2}{a}\int_{0}^{a}f(x)\sin\frac{(2n-1)\pi x}{2a}dx,\quad u(x,y)=-\frac{2000}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\left[(-1)^{n}+\frac{4}{(2n-1)\pi }\right]e^{-(2n-1)\pi y/10}\sin\frac{(2n-1)\pi x}{10}\)

    35. \(u(x,y)= \sum_{n=1}^{\infty}\frac{A_{n}\sinh n\pi (b-y)/a+B_{n}\sinh n\pi y/a}{\sinh n\pi b/a}\sin\frac{n\pi x}{a}+ \sum_{n=1}^{\infty}\frac{C_{n}\sinh n\pi (a-x)/b+D_{n}\sinh n\pi x/b}{\sinh n\pi a/b}\sin\frac{n\pi y}{b}\)

    36. \(u(x,y)=C+\frac{a}{\pi } \sum_{n=1}^{\infty}\frac{B_{n}\cosh n\pi y/a-A_{n}\cosh n\pi (y-b)/a}{n\sinh n\pi b/a}\cos\frac{n\pi x}{a}+\frac{b}{\pi } \sum_{n=1}^{\infty}\frac{D_{n}\cosh n\pi x/b-C_{n}\cosh n\pi (x-a)/b}{n\sinh n\pi a/b}\cos\frac{n\pi y}{b}\)