91Ó°ÊÓ

Solve the boundary value problem. $$ \begin{aligned} &u_{t}=u_{x x}, 00 ; u(0, t)=u(\pi, t)=0, \\ &u(x, 0)=4 \sin 4 x \cos 2 x \end{aligned} $$

Sketch the graph of the function \(f\) defined for all \(t\) by the given formula, and determine whether it is periodic. If so, find its smallest period. $$ f(t)=\sin \frac{\pi t}{3} $$

Consider the boundary value problem $$ \begin{aligned} u_{x x}+u_{y y} &=0 \\ u_{x}(0, y) &=u_{x}(a, y)=u(x, 0)=0, \\ u(x, b) &=f(x) \end{aligned} $$ corresponding to a rectangular plate \(0

Solve the boundary value problems in Problems 1 through \(10 .\) \(4 y_{t t}=y_{x x}, 00 ; y(0, t)=y(2, t)=0\), \(y(x, 0)=\frac{1}{5} \sin \pi x \cos \pi x, y_{t}(x, 0)=0\)

Find the steady periodic solution \(x_{s p}(t)\) of each of the differential equations. Use a computer algebra system to plot enough terms of the series to determine the visual appearance of the graph of \(x_{s p}(t) .\) \(x^{\prime \prime}+10 x=F(t)\), where \(F(t)\) is the odd function of period 2 such that \(F(t)=t-t^{2}\) if \(0

In Problems, a function \(f(t)\) defined on an interval \(0

Solve the boundary value problems in Problems 1 through \(10 .\) \(y_{t t}=25 y_{x x}, 00 ; y(0, t)=y(3, t)=0\), \(y(x, 0)=\frac{1}{4} \sin \pi x, y_{t}(x, 0)=10 \sin 2 \pi x\)

Sketch the graph of the function \(f\) defined for all \(t\) by the given formula, and determine whether it is periodic. If so, find its smallest period. $$ f(t)=\tan t $$

Solve the boundary value problem. $$ \begin{aligned} &u_{t}=2 u_{x x}, 00 ; u_{x}(0, t)=u_{x}(3, t)=0, \\ &u(x, 0)=4 \cos \frac{2}{3} \pi x-2 \cos \frac{4}{3} \pi x \end{aligned} $$

In Problems, a function \(f(t)\) defined on an interval \(0