Homework Ten

96

$$u(x,t)=\frac{2 A l}{\pi^2} \sum\limits_{n=1}^\infty \frac{\s... ...n \pi}{4}}{n^2} \sin\frac{n\pi x}{l}\cos\frac{n\pi\alpha t}{l}.$$

97

$$u(x,t) = \frac{4 l^2 A}{\pi^3 a} \sum\limits_{n=1}^\infty \sin\frac{n\pi}{2}\sin\frac{n\pi x}{l}\sin\frac{n \pi \alpha t}{l}.$$

98

$$u(x,t) = \sum\limits_{n=1}^\infty c_n \sin \frac{(2 n -1)\pi x}{2 l} \cos\frac{(2 n-1)\pi\alpha t}{2 l},$$

$$c_n = \frac{2}{l}\int\limits_0^l f(x) \sin\frac{(2 n - 1)\pi x}{2 l} d x.$$

99

$$u(x,t)=\sum\limits_{n=1}^\infty c_n \sin\frac{n\pi x}{l} \cos\left(\sqrt{\alpha^2 + \frac{n^2\pi^2}{l^2}}t\right),$$

$$c_n = \frac{2}{l}\int\limits_0^l f(x) \sin \frac{n\pi x}{l} d x.$$

100

$$T''(t)+\alpha^2\lambda^2 T(t) =0,$$

$$r^2 R''(r) + r R'(r) + (\lambda^2 r^2 - \mu^2) R(r) = 0,$$

$$\Theta '' + \mu^2 \Theta = 0.$$