91Ó°ÊÓ

Determine whether the method of separation of variables can be used to replace the given partial differential equation by a pair of ordinary differential equations. If so, find the equations. $$ t u_{x x}+x u_{t}=0 $$

Determine whether the given function is periodic. If so, find its fundamental period. $$ \cos 2 \pi x $$

Find the solution \(u(x, y)\) of Laplace's equation in the rectangle \(0

Either solve the given boundary value problem or else show that it has no solution. \(y^{\prime \prime}+2 y=0, \quad y^{\prime}(0)=1, \quad y^{\prime}(\pi)=0\)

Determine whether the method of separation of variables can be used to replace the given partial differential equation by a pair of ordinary differential equations. If so, find the equations. $$ u_{x x}+u_{x t}+u_{t}=0 $$

find the steady-state solution of the heat conduction equation \(\alpha^{2} u_{x x}=u_{t}\) that satisfies the given set of boundary conditions. $$ u_{x}(0, t)=0, \quad u(I, t)=0 $$

Determine whether the given function is even, odd, or neither. $$ \tan 2 x $$

Carry out the following steps. Let \(L=10\) and \(a=1\) in parts (b) through (d). (a) Find the displacement \(u(x, t)\) for the given initial position \(f(x) .\) (b) Plot \(u(x, t)\) versus \(x\) for \(0 \leq x \leq 10\) and for several values of \(t\) between \(t=0\) and \(t=20\). (c) Plot \(u(x, t)\) versus \(t\) for \(0 \leq t \leq 20\) and for several values of \(x .\) (d) Construct an animation of the solution in time for at least one period. (e) Describe the motion of the string in a few sentences. \(f(x)=8 x(L-x)^{2} / L^{3}\)

Determine whether the given function is periodic. If so, find its fundamental period. $$ \sinh 2 x $$

(a) Find the solution \(u(x, y)\) of Laplace's equation in the rectangle \(0