91Ó°ÊÓ

Use d'Alembert's formula (4) to solve the boundary value problem (1) - (3) for a string of unit length, subject to the given conditions. In each case, describe completely \(f^{*}\) and \(G\) (an antiderivative of \(\left.g^{\circ}\right)\) (see Examples 2 and 3 for hints). $$ f(x)=\sin \pi x \cos \pi x, g(x)=0, \quad c=\frac{1}{\pi} $$

Use d'Alembert's formula (4) to solve the boundary value problem (1) - (3) for a string of unit length, subject to the given conditions. In each case, describe completely \(f^{*}\) and \(G\) (an antiderivative of \(\left.g^{\circ}\right)\) (see Examples 2 and 3 for hints). $$ f(x)=0, g(x)=1, c=1 $$

Damped vibrations of a string. In the presence of resistance proportional to velocity, the one dimensional wave equation becomes $$ \frac{\partial^{2} u}{\partial t^{2}}+2 k \frac{\partial u}{\partial t}=c^{2} \frac{\partial^{2} u}{\partial x^{2}} $$

Solve the heat equation (11) in a unit square \((a=b=1)\) with the given initial temperature distribution \(f\). Assume that the edyes are kept at zero temperature and that \(c=1\). $$ f(x, y)=\sin \pi z \sin \pi y $$