91Ó°ÊÓ

\(f(x)=1+x\) \(0<\) \(x<\) \(\pi\)

Find a solution to the following Dirichlet problem for an exterior domain: \(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}=0, \quad 1 < r, \quad-\pi \leq \theta \leq \pi,\) \(u(1, \theta)=f(\theta) \quad-\pi \leq \theta \leq \pi\) \(u(r, \theta) \quad\) remains bounded as \(r \rightarrow \infty\)

$$\begin{array} { l } { y ^ { \prime \prime } + \lambda y = 0 ; \quad 0 < x < \pi } \\ { y ( 0 ) - y ^ { \prime } ( 0 ) = 0 , \quad y ( \pi ) = 0 } \end{array}$$

Find a formal solution to the initial-boundary value problem $$\frac{\partial u}{\partial t}=2 \frac{\partial^{2} u}{\partial x^{2}}+4 x, 0<\mathcal{x}<\pi,t>0,$$ $$u(0, t)=u(\pi, t)=0, \quad t>0,$$ $$u(x, 0)=\sin x,0<\mathcal{x}<\pi$$

\(f(x)=e^{x}\) \(0<\) \(x<\) 1

$$f(x)=x^{2}$$ $$-1< x<1$$

\(f(x)=e^{-x}\) \(0<\) \(x<\) 1

Find a solution to the following Neumann problem for an exterior domain: \(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}=0, \quad 1< r, \quad-\pi \leq \theta \leq \pi,\) \(\frac{\partial u}{\partial r}(1, \theta)=f(\theta), \quad-\pi \leq \theta \leq \pi,\) \(u(r, \theta) \quad\) remains bounded as \(r \rightarrow \infty\)

$$ f(x)=x^{2}, \quad g(x)=0 $$

$$\begin{array} { l } { y ^ { \prime \prime } - 2 y ^ { \prime } + \lambda y = 0 ; \quad 0 < x < \pi } \\ { y ( 0 ) = 0 , \quad y ( \pi ) = 0 } \end{array}$$