# 2.8: Problems

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## Exercise $$\PageIndex{1}$$

Solve the following initial value problems.

1. $$x''+x=0,\quad x(0)=2,\quad x'(0)=0$$.
2. $$y''+2y'-8y=0,\quad y(0)=1,\quad y'(0)=2$$.
3. $$x^2y''-2xy'-4y=0,\quad y(1)=1,\quad y'(1)=0$$.

## Exercise $$\PageIndex{2}$$

Solve the following boundary value problems directly, when possible.

1. $$x''+x=2,\quad x(0)=0,\quad x'(1)=0$$.
2. $$y''+2y'-8y=0,\quad y(0)=1,\quad y(1)=0$$.
3. $$y''+y=0,\quad y(0)=1,\quad y(\pi )=0$$.

## Exercise $$\PageIndex{3}$$

Consider the boundary value problem for the deflection of a horizontal beam fixed at one end,

$\frac{d^4y}{dx^4}=C,\quad y(0)=0,\quad y'(0)=0,\quad y''(L)=0,\quad y'''(L)=0.\nonumber$

Solve this problem assuming that $$C$$ is a constant.

## Exercise $$\PageIndex{4}$$

Find the product solutions, $$u(x, t) = T(t)X(x)$$, to the heat equation, $$u_t − u_{xx} = 0$$, on $$[0, π]$$ satisfying the boundary conditions $$u_x(0, t) = 0$$ and $$u(π, t) = 0$$.

## Exercise $$\PageIndex{5}$$

Find the product solutions, $$u(x, t) = T(t)X(x)$$, to the wave equation $$u_{tt} = 2u_{xx}$$, on $$[0, 2π]$$ satisfying the boundary conditions $$u(0, t) = 0$$ and $$u_x(2π, t) = 0$$.

## Exercise $$\PageIndex{6}$$

Find product solutions, $$u(x, t) = X(x)Y(y)$$, to Laplace’s equation, $$u_{xx} + u_{yy} = 0$$, on the unit square satisfying the boundary conditions $$u(0, y) = 0$$, $$u(1, y) = g(y)$$, $$u(x, 0) = 0$$, and $$u(x, 1) = 0$$.

## Exercise $$\PageIndex{7}$$

Consider the following boundary value problems. Determine the eigenvalues, $$λ$$, and eigenfunctions, $$y(x)$$ for each problem.

1. $$y'' + λy = 0,\quad y(0) = 0,\quad y'(1) = 0$$.
2. $$y'' − λy = 0,\quad y(−π) = 0,\quad y'(π) = 0$$.
3. $$x^2y'' + xy' + λy = 0,\quad y(1) = 0,\quad y(2) = 0$$.
4. $$(x^2y')'+\lambda y=0,\quad y(1)=0,\quad y'(e)=0$$.

## Note

In problem d you will not get exact eigenvalues. Show that you obtain a transcendental equation for the eigenvalues in the form $$\tan z = 2z$$. Find the first three eigenvalues numerically.

## Exercise $$\PageIndex{8}$$

Classify the following equations as either hyperbolic, parabolic, or elliptic.

1. $$u_{yy} + u_{xy} + u_{xx} = 0$$.
2. $$3u_{xx} + 2u_{xy} + 5u_{yy} = 0$$.
3. $$x^2u_{xx}+2xyu_{xy}+y^2u_{yy}=0$$.
4. $$y^2u_{xx}+2xyu_{xy}+(x^2+4x^4)u_{yy}=0$$.

## Exercise $$\PageIndex{9}$$

Use d’Alembert’s solution to prove

$f(−ζ) = f(ζ),\quad g(−ζ) = g(ζ)\nonumber$

for the semi-infinite string satisfying the free end condition $$u_x(0, t) = 0$$.

## Exercise $$\PageIndex{10}$$

Derive a solution similar to d’Alembert’s solution for the equation $$u_{tt} + 2u_{xt} − 3u = 0$$.

## Exercise $$\PageIndex{11}$$

Construct the appropriate periodic extension of the plucked string initial profile given by

$f(x)=\left\{\begin{array}{cc}x,&0\leq x\leq\frac{\ell}{2}, \\ \ell -x,&\frac{\ell}{2}\leq x\leq\ell ,\end{array}\right.\nonumber$

satisfying the boundary conditions at $$u(0, t) = 0$$ and $$u_x(\ell, t) = 0$$ for $$t > 0$$.

## Exercise $$\PageIndex{12}$$

Find and sketch the solution of the problem

\begin{aligned} u_{tt}&=u_{xx},\quad 0\leq x\leq 1,\quad t> o \\ u(x,0)&=\left\{\begin{array}{rr}0,&0\leq x<\frac{1}{4}, \\ 1,&\frac{1}{4}\leq x\leq\frac{3}{4}, \\ 0,&\frac{3}{4}<x\leq 1,\end{array}\right. \\ u_t(x,0)&=0, \\ u(0,t)&=0,\quad t>0, \\ u(1,t)&=0,\quad t>0,\end{aligned}

This page titled 2.8: Problems is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Russell Herman via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.