For the given $f(x)$, find the solution to the heat flow problem.

$$\begin{aligned} \frac{\partial u}{\partial t} &=5 \frac{\partial^2 u}{\partial x^2}, \quad 0<x<\pi, \quad t>0, \\ u(0, t) &=u(\pi, t)=0, \quad t>0, \\ u(x, 0) &=f(x), \quad 0<x<\pi, \end{aligned}$$

$f(x)=1-\cos 2 x$

Compute the Fourier series for the given function $f$ on the specified interval. Use a computer or graphing calculator to plot a few partial sums of the Fourier series. \

$$f(x)=\left\{\begin{aligned} 0, &-\pi &<x<-\pi / 2 \\-1, &-\pi / 2 &<x<0, \\ 1, & & 0 &<x<\pi / 2, \\ 0, & \pi / 2 &<x<\pi \end{aligned}\right.$$

Solve the heat flow problem $(1)$-$(3)$ with $\beta=3, L=\pi$, and the given function $f(x)$. \

$$f(x)=\sin 3 x+5 \sin 7 x-2 \sin 13 x$$

Find the solution to the initial value problem.

$$\begin{aligned} &\frac{\partial^2 u}{\partial t^2}=\alpha^2 \frac{\partial^2 u}{\partial x^2}, \quad-\infty<x<\infty, \quad t>0, \\ &u(x, 0)=f(x), \quad-\infty<x<\infty, \\ &\frac{\partial u}{\partial t}(x, 0)=g(x), \quad-\infty<x<\infty, \end{aligned}$$

for the given functions $f(x)$ and $g(x)$. \

$$f(x)=\sin 3 x, \quad g(x) \equiv 1$$

Solve the heat flow problem $(1)$-$(3)$ with $\beta=3, L=\pi$, and the given function $f(x)$. \

$$f(x)=\sin x-6 \sin 4 x$$

Find the solution to the initial value problem.

$$\begin{aligned} &\frac{\partial^2 u}{\partial t^2}=\alpha^2 \frac{\partial^2 u}{\partial x^2}, \quad-\infty<x<\infty, \quad t>0, \\ &u(x, 0)=f(x), \quad-\infty<x<\infty, \\ &\frac{\partial u}{\partial t}(x, 0)=g(x), \quad-\infty<x<\infty, \end{aligned}$$

for the given functions $f(x)$ and $g(x)$. \

$$f(x)=x, \quad g(x)=x$$

Solve the heat flow problem $(1)$-$(3)$ with $\beta=3, L=\pi$, and the given function $f(x)$. \

$$f(x)=\sin x-6 \sin 4 x$$

Find a formal solution to the initialboundary value problem.

$$\begin{aligned} &\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}, \quad 0<x<\pi, \quad 0<y<\pi, \quad t>0 \\ &\frac{\partial u}{\partial x}(0, y, t)=\frac{\partial u}{\partial x}(\pi, y, t)=0, \quad 0<y<\pi, \quad t>0 \\ &u(x, 0, t)=u(x, \pi, t)=0, \quad 0<x<\pi, \quad t>0 \\ &u(x, y, 0)=f(x, y), \quad 0<x<\pi, \quad 0<y<\pi \end{aligned}$$

for the given function $f(x, y)$. \

$$f(x, y)=\cos 6 x \sin 4 y-3 \cos x \sin 11 y$$

Compute the Fourier series for the given function $f$ on the specified interval. Use a computer or graphing calculator to plot a few partial sums of the Fourier series. \

$$f(x)=e^x, \quad-\pi<x<\pi$$

Find a solution to the following Dirichlet problem for a half disk:

$$\begin{aligned} &\frac{\partial^2 u}{\partial r^2}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2}=0, \quad 0<r<1, \quad 0<\theta<\pi \\ &u(r, 0)=0, \quad 0 \leq r \leq 1 \\ &u(r, \pi)=0, \quad 0 \leq r \leq 1 \\ &u(1, \theta)=\sin 3 \theta, \quad 0 \leq \theta \leq \pi \\ &u(0, \theta) \quad \text { bounded. } \end{aligned}$$

Compute the Fourier cosine series for the given function.
$f(x)=\sin x, \quad 0<x<\pi$

Find a formal solution to the initial-boundary value problem

$$\begin{aligned} &\frac{\partial u}{\partial t}=3 \frac{\partial^2 u}{\partial x^2}+5, \quad 0<x<\pi, \quad t>0 \\ &u(0, t)=u(\pi, t)=1, \quad t>0 \\ &u(x, 0)=1, \quad 0<x<\pi \end{aligned}$$

Compute the Fourier series for the given function $f$ on the specified interval. Use a computer or graphing calculator to plot a few partial sums of the Fourier series. \

$$f(x)= \begin{cases}x, & 0<x<\pi, \\ x+\pi, & -\pi<x<0\end{cases}$$

Find the solution to the initial value problem.

$$\begin{aligned} &\frac{\partial^2 u}{\partial t^2}=\alpha^2 \frac{\partial^2 u}{\partial x^2}, \quad-\infty<x<\infty, \quad t>0, \\ &u(x, 0)=f(x), \quad-\infty<x<\infty, \\ &\frac{\partial u}{\partial t}(x, 0)=g(x), \quad-\infty<x<\infty, \end{aligned}$$

for the given functions $f(x)$ and $g(x)$.
$f(x)=x^2, \quad g(x) \equiv 0$

Find the values of $\lambda$ (eigenvalues) for which the given problem has a nontrivial solution. Also determine the corresponding nontrivial solutions (eigenfunctions).
$y^{\prime \prime}-2 y^{\prime}+\lambda y=0 ; \quad 0<x<\pi,\\ y(0)=0, \quad y(\pi)=0$