91Ó°ÊÓ

Find the deflection \(u(x, y, t)\) of the square membrane of side \(\pi\) and \(c^{2}=1\) if the initial velocity is 0 and the initial deflection is $$0.1 x y(\pi-x)(\pi-y)$$

Find the potential in the interior of the sphere \(S: r=R=1\) if this interior is free of charges and the potential on \(S\) is: $$f(\phi)=\cos \phi$$

Find the electrostatic potencial in the semidisk \(r<1,0<\theta<\pi\) which equals \(110 \theta(\pi-\theta)\) on the semicircle \(r=1\) and 0 on the segment \(-1

Find the potential in the interior of the sphere \(S: r=R=1\) if this interior is free of charges and the potential on \(S\) is: $$f(\phi)=\cos 3 \phi$$

Find the potential in the interior of the sphere \(S: r=R=1\) if this interior is free of charges and the potential on \(S\) is: $$f(\phi)=\sin ^{2} \phi$$

Find the potential in the interior of the sphere \(S: r=R=1\) if this interior is free of charges and the potential on \(S\) is: $$f(\phi)=35 \cos 4 \phi+20 \cos 2 \phi+9$$

Represent \(f(x, y)\) \((0

Represent \(f(x, y)\) \((0

Represent \(f(x, y)\) \((0

The boundary condition of heat transfer $$-u_{n}(\pi, t)=k\left[u(\pi, t)-u_{0}\right]$$ applies when a bar of length \(\pi\) with \(c=1\) is laterally insulated, the left end \(x=0\) is kept at \(0^{\circ} \mathrm{C}\), and at the right end heat is flowing into air of constant temperature \(u_{0}\), Let \(k=1\) for simplicity, and \(u_{0}=0\). Show that a solution is \(u(x, t)=\sin p x e^{-y^{2} t}\), where \(p\) is a solution of \(\tan p \pi=-p .\) Show graphically that this equation has infinitely many positive solutions \(p_{1}, p_{2}, p_{3}, \cdots^{3},\) where \(p_{x}>n-\frac{1}{2}\) and \(\lim _{h \rightarrow \infty}\left(p_{n}-n+\frac{1}{2}\right)=0 .\) (Formula (19) is also known as radiation boundary condition, but this is misleading; see Ref. \(\left[\mathrm{C}^{3}\right] .\) p. 19.)