2.8: Problems
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- 90923
Solve the following initial value problems.
- \(x''+x=0,\quad x(0)=2,\quad x'(0)=0\).
- \(y''+2y'-8y=0,\quad y(0)=1,\quad y'(0)=2\).
- \(x^2y''-2xy'-4y=0,\quad y(1)=1,\quad y'(1)=0\).
Solve the following boundary value problems directly, when possible.
- \(x''+x=2,\quad x(0)=0,\quad x'(1)=0\).
- \(y''+2y'-8y=0,\quad y(0)=1,\quad y(1)=0\).
- \(y''+y=0,\quad y(0)=1,\quad y(\pi )=0\).
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.
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 \(u(π, t) = 0\).
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 \(u_x(2π, t) = 0\).
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 \(u(x, 1) = 0\).
Consider the following boundary value problems. Determine the eigenvalues, \(λ\), and eigenfunctions, \(y(x)\) for each problem.
- \(y'' + λy = 0,\quad y(0) = 0,\quad y'(1) = 0\).
- \(y'' − λy = 0,\quad y(−π) = 0,\quad y'(π) = 0\).
- \(x^2y'' + xy' + λy = 0,\quad y(1) = 0,\quad y(2) = 0\).
- \((x^2y')'+\lambda y=0,\quad y(1)=0,\quad y'(e)=0\).
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.
Classify the following equations as either hyperbolic, parabolic, or elliptic.
- \(u_{yy} + u_{xy} + u_{xx} = 0\).
- \(3u_{xx} + 2u_{xy} + 5u_{yy} = 0\).
- \(x^2u_{xx}+2xyu_{xy}+y^2u_{yy}=0\).
- \(y^2u_{xx}+2xyu_{xy}+(x^2+4x^4)u_{yy}=0\).
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 \(u_x(0, t) = 0\).
Derive a solution similar to d’Alembert’s solution for the equation \(u_{tt} + 2u_{xt} − 3u = 0\).
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 \(t > 0\).
Find and sketch the solution of the problem
\[\begin{aligned} u_{tt}&=u_{xx},\quad 0\leq x\leq 1,\quad t> o \\ u(x,0)&=\left\{\begin{array}{rr}0,&0\leq x<\frac{1}{4}, \\ 1,&\frac{1}{4}\leq x\leq\frac{3}{4}, \\ 0,&\frac{3}{4}<x\leq 1,\end{array}\right. \\ u_t(x,0)&=0, \\ u(0,t)&=0,\quad t>0, \\ u(1,t)&=0,\quad t>0,\end{aligned}\]