91Ó°ÊÓ

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. $$ f(t)=\cos ^{2} 3 t $$

Solve the boundary value problem. $$ \begin{aligned} &5 u_{t}=u_{x x}, 00 ; u(0, t)=u(10, t)=0, \\ &u(x, 0)=4 x \end{aligned} $$

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

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

The mass \(m\) and Hooke's constant \(k\) for a mass-and-spring system are given. Determine whether or not pure resonance will occur under the influence of the given external periodic force \(F(t) .\) \(m=1, k=4 \pi^{2} ; F(t)\) is the odd function of period 2 with \(F(t)=2 t\) for \(0

In Problems 11 through 26, the values of a period \(2 \pi\) function \(f(t)\) in one full period are given. Sketch several periods of its graph and find its Fourier series. $$ f(t) \equiv 1,-\pi \leqq t \leqq \pi $$

Solve the boundary value problem. $$ \begin{aligned} &5 u_{t}=u_{x x}, 00 ; u_{x}(0, t)=u_{x}(10, t)=0, \\ &u(x, 0)=4 x \end{aligned} $$

Find formal Fourier series solutions of the endpoint value problems in Problems. $$ x^{\prime \prime}+2 x=1, x(0)=x(\pi)=0 $$

The mass \(m\) and Hooke's constant \(k\) for a mass-and-spring system are given. Determine whether or not pure resonance will occur under the influence of the given external periodic force \(F(t) .\) \(m=3, k=48 ; F(t)\) is the even function of period \(2 \pi\) with \(F(t)=t\) for \(0

The values of a periodic function \(f(t)\) in one full period are given; at each discontinuity the value of \(f(t)\) is that given by the average value condition in Sketch the graph of \(f\) and find its Fourier series. $$ f(t)=\cos \frac{\pi t}{2},-1