Chapter 13

Solve the heat equation \(k u_{x x}=u_{t}, 00\) subject to the given conditions. $$ \begin{aligned} &u(0, t)=100, \quad u(1, t)=100 \\ &u(x, 0)=0 \end{aligned} $$

In Problems \(1-10\), solve Laplace's equation (1) for a rectangular plate subject to the given boundary conditions. \(u(0, y)=0, u(a, y)=0\) \(u(x, 0)=0, u(x, b)=f(x)\)

In Problems \(1-6\), solve the wave equation (1) subject to the given conditions. \(u(0, t)=0, \quad u(L, t)=0, \quad t>0\) \(u(x, 0)=\frac{1}{4} x(L-x),\left.\frac{\partial u}{\partial t}\right|_{t=0}=0, \quad 0

A rod of length \(L\) coincides with the interval \([0, L]\) on the \(x\) -axis. Set up the boundary-value problem for the temperature \(u(x, t)\) The left end is held at temperature zero, and the right end is insulated. The initial temperature is \(f(x)\) throughout.

In Problems, solve the heat equation (1) subject to the given conditions. Assume a rod of length \(L\). $$ \begin{aligned} &u(0, t)=0, u(L, t)=0 \\ &u(x, 0)=\left\\{\begin{array}{lr} 1, & 0

In Problems 1-16, use separation of variables to find, if possible, product solutions for the given partial differential equation. $$ \frac{\partial u}{\partial x}=\frac{\partial u}{\partial y} $$

Solve Laplace's equation (1) for a rectangular plate subject to the given boundary conditions.\(u(0, y)=0, u(a, y)=0\) \(u(x, 0)=0, u(x, b)=f(x)\)

Solve the wave equation (1) subject to the given conditions.\(u(0, t)=0, \quad u(L, t)=0, \quad t>0\) \(u(x, 0)=\frac{1}{4} x(L-x),\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=0, \quad 0

Solve the heat equation (1) subject to the given conditions. Assume a rod of length \(L\).$$ \begin{aligned} &u(0, t)=0, \quad u(L, t)=0 \\ &u(x, 0)=\left\\{\begin{array}{lr} 1, & 0

Use separation of variables to find, if possible, product solutions for the given partial differential equation.\(\frac{\partial u}{\partial x}=\frac{\partial u}{\partial y}\)

Solve the boundary-value problem $$ \begin{aligned} &k \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t}, 00 \\ &u(0, t)=0,\left.\frac{\partial u}{\partial x}\right|_{x=1}=-h\left(u(1, t)-u_{0}\right), h>0, t>0 \\ &u(x, 0)=f(x), 0

In Problems \(1-10\), solve Laplace's equation (1) for a rectangular plate subject to the given boundary conditions. \(u(0, y)=0, u(a, y)=0\) \(\left.\frac{\partial u}{\partial y}\right|_{y=0}=0, u(x, b)=f(x)\)

In Problems, solve the heat equation (1) subject to the given conditions. Assume a rod of length \(L\). $$ \begin{aligned} &u(0, t)=0, \quad u(L, t)=0 \\ &u(x, 0)=x(L-x) \end{aligned} $$

Use separation of variables to find, if possible, product solutions for the given partial differential equation. $$ \frac{\partial u}{\partial x}+3 \frac{\partial u}{\partial y}=0 $$

Solve Laplace's equation (1) for a rectangular plate subject to the given boundary conditions.\(u(0, y)=0, u(a, y)=0\) \(\left.\frac{\partial u}{\partial y}\right|_{y=0}=0, u(x, b)=f(x)\)

Solve the wave equation (1) subject to the given conditions.\(u(0, t)=0, \quad u(L, t)=0, \quad t>0\) \(u(x, 0)=0,\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=x(L-x), \quad 0

Solve the heat equation (1) subject to the given conditions. Assume a rod of length \(L\).$$ \begin{aligned} &u(0, t)=0, u(L, t)=0 \\ &u(x, 0)=x(L-x) \end{aligned} $$

Find the steady-state temperature for a rectangular plate for which the boundary conditions are $$ \begin{aligned} &u(0, y)=0,\left.\frac{\partial u}{\partial x}\right|_{x=a}=-h u(a, y), h>0,0

A rod of length \(L\) coincides with the interval \([0, L]\) on the \(x\) -axis. Set up the boundary-value problem for the temperature \(u(x, t)\) The left end is held at temperature \(100^{\circ}\), and there is heat transfer from the right end into the surrounding medium at temperature zero. The initial temperature is \(f(x)\) throughout.

Use separation of variables to find, if possible, product solutions for the given partial differential equation. $$ u_{x}+u_{y}=u $$

Solve Laplace's equation (1) for a rectangular plate subject to the given boundary conditions.\(u(0, y)=0, u(a, y)=0\) \(u(x, 0)=f(x), u(x, b)=0\)

Find a steady-state solution \(\psi(x)\) of the boundary-value problem $$ \begin{aligned} &k \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t}, 00 \\ &u(0, t)=u_{0}, \quad-\left.\frac{\partial u}{\partial x}\right|_{x=\pi}=u(\pi, t)-u_{1}, t>0 \\ &u(x, 0)=0,0

Solve the boundary-value problem $$ \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0, x>0,00, x>0 \end{aligned} $$

In Problems \(1-10\), solve Laplace's equation (1) for a rectangular plate subject to the given boundary conditions. \(\left.\frac{\partial u}{\partial x}\right|_{x=0}=0,\left.\frac{\partial u}{\partial x}\right|_{x=a}=0\) \(u(x, 0)=x, u(x, b)=0\)

A rod of length \(L\) coincides with the interval \([0, L]\) on the \(x\) -axis. Set up the boundary-value problem for the temperature \(u(x, t)\) There is heat transfer from the left end into a surrounding medium at temperature \(20^{\circ}\), and the right end is insulated. The initial temperature is \(f(x)\) throughout.

Solve Laplace's equation (1) for a rectangular plate subject to the given boundary conditions.\(\left.\frac{\partial u}{\partial x}\right|_{x=0}=0,\left.\frac{\partial u}{\partial x}\right|_{x=a}=0\) \(u(x, 0)=x, u(x, b)=0\)

Use separation of variables to find, if possible, product solutions for the given partial differential equation.\(u_{x}=u_{y}+u\)

Solve the boundary-value problem $$ \begin{aligned} &k \frac{\partial^{2} u}{\partial x^{2}}+A e^{-\beta x}=\frac{\partial u}{\partial t}, \beta>0,00 \\ &u(0, t)=0, \quad u(1, t)=0, t>0 \\ &u(x, 0)=f(x), 0

Find the temperature \(u(x, t)\) in a rod of length \(L\) if the initial temperature is \(f(x)\) throughout and if the end \(x=0\) is maintained at temperature zero and the end \(x=L\) is insulated.

Solve the wave equation (1) subject to the given conditions. \(u(0, t)=0, u(1, t)=0, t>0\) \(u(x, 0)=x(1-x),\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=x(1-x), \quad 0

In Problems \(1-10\), solve Laplace's equation (1) for a rectangular plate subject to the given boundary conditions. \(u(0, y)=0, u(1, y)=1-y\) \(\left.\frac{\partial u}{\partial y}\right|_{y=0}=0,\left.\frac{\partial u}{\partial y}\right|_{y=1}=0\)

Use separation of variables to find, if possible, product solutions for the given partial differential equation. $$ x \frac{\partial u}{\partial x}=y \frac{\partial u}{\partial y} $$

Solve the boundary-value problem $$ \begin{aligned} &k \frac{\partial^{2} u}{\partial x^{2}}-h u=\frac{\partial u}{\partial t}, 00 \\ &u(0, t)=0, \quad u(\pi, t)=u_{0}, t>0 \\ &u(x, 0)=0,0

A rod of length \(L\) coincides with the interval \([0, L]\) on the \(x\) -axis. Set up the boundary-value problem for the temperature \(u(x, t)\) The ends are insulated, and there is heat transfer from the lateral surface of the rod into the surrounding medium held at temperature \(50^{\circ} .\) The initial temperature is \(100^{\circ}\) throughout.

Use separation of variables to find, if possible, product solutions for the given partial differential equation. $$ y \frac{\partial u}{\partial x}+x \frac{\partial u}{\partial y}=0 $$

The partial differential equation $$ \frac{\partial^{2} u}{\partial x^{2}}+x^{2}=\frac{\partial^{2} u}{\partial t^{2}} $$ is a form of the wave equation when an external vertical force proportional to the square of the horizontal distance from the left end is applied to the string. The string is secured at \(x=0\) one unit above the \(x\)-axis and on the \(x\)-axis at \(x=1\) for \(t>0\). Find the displacement \(u(x, t)\) if the string starts from rest from the initial displacement \(f(x)\).

Solve the boundary-value problem $$ \begin{aligned} &a^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}, 00 \\ &u(0, t)=0,\left.E \frac{\partial u}{\partial x}\right|_{x=L}=F_{0}, t>0 \\ &u(x, 0)=0,\left.\frac{\partial u}{\partial t}\right|_{t=0}=g(x), 0

Solve the boundary-value problem $$ \begin{aligned} &\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0,0

In Problems \(1-10\), solve Laplace's equation (1) for a rectangular plate subject to the given boundary conditions. \(\left.\frac{\partial u}{\partial x}\right|_{x=0}=u(0, y), u(\pi, y)=1\) \(u(x, 0)=0, u(x, \pi)=0\)

A string of length \(L\) coincides with the interval \([0, L]\) on the \(x\) -axis. Set up the boundary-value problem for the displacement \(u(x, t)\). The ends are secured to the \(x\) -axis. The string is released from rest from the initial displacement \(x(L-x)\).

A thin wire coinciding with the \(x\) -axis on the interval \([-L, L]\) is bent into the shape of a circle so that the ends \(x=-L\) and \(x=L\) are joined. Under certain conditions the temperature \(u(x, t)\) in the wire satisfies the boundary-value problem $$ \begin{aligned} &k_{\partial x^{2}}^{\partial^{2} u}=\frac{\partial u}{\partial t},-L0 \\\ &u(-L, t)=u(L, t), t>0 \\ &\left.\frac{\partial u}{\partial x}\right|_{x=-L}=\left.\frac{\partial u}{\partial x}\right|_{x=L}, t>0 \\ &u(x, 0)=f(x),-L

Use separation of variables to find, if possible, product solutions for the given partial differential equation. $$ \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial x \partial y}+\frac{\partial^{2} u}{\partial y^{2}}=0 $$

Solve Laplace's equation (1) for a rectangular plate subject to the given boundary conditions.\(\left.\frac{\partial u}{\partial x}\right|_{x=0}=u(0, y), u(\pi, y)=1\) \(u(x, 0)=0, u(x, \pi)=0\)

The initial temperature in a rod of unit length is \(f(x)\) throughout. There is heat transfer from both ends, \(x=0\) and \(x=1\), into a surrounding medium kept at a constant temperature zero. Show that $$ u(x, t)=\sum_{n=1}^{\infty} A_{n} e^{-k \alpha_{n}^{2} t}\left(\alpha_{n} \cos \alpha_{n} x+h \sin \alpha_{n} x\right) $$ where $$ A_{n}=\frac{2}{\left(\alpha_{n}^{2}+2 h+h^{2}\right)} \int_{0}^{1} f(x)\left(\alpha_{n} \cos \alpha_{n} x+h \sin \alpha_{n} x\right) d x $$ The eigenvalues are \(\lambda_{n}=\alpha_{n}^{2}, n=1,2,3, \ldots\), where the \(\alpha_{n}\) are the consecutive positive roots of \(\tan \alpha=2 \alpha h /\left(\alpha^{2}-h^{2}\right)\)

In Problems \(1-10\), solve Laplace's equation (1) for a rectangular plate subject to the given boundary conditions. \(u(0, y)=0, u(1, y)=0\) \(\left.\frac{\partial u}{\partial y}\right|_{y=0}=u(x, 0), u(x, 1)=f(x)\)

A string of length \(L\) coincides with the interval \([0, L]\) on the \(x\) -axis. Set up the boundary-value problem for the displacement \(u(x, t)\). The ends are secured to the \(x\) -axis. Initially the string is undisplaced but has the initial velocity \(\sin (\pi x / L)\).

Use separation of variables to find, if possible, product solutions for the given partial differential equation. $$ y \frac{\partial^{2} u}{\partial x \partial y}+u=0 $$

Solve Laplace's equation (1) for a rectangular plate subject to the given boundary conditions.\(u(0, y)=0, u(1, y)=0\) \(\left.\frac{\partial u}{\partial y}\right|_{y=0}=u(x, 0), u(x, 1)=f(x)\)

When a vibrating string is subjected to an external vertical force that varies with the horizontal distance from the left end, the wave equation takes on the form $$ a^{2} \frac{\partial^{2} u}{\partial x^{2}}+A x=\frac{\partial^{2} u}{\partial t^{2}} $$ where \(A\) is constant. Solve this partial differential equation subject to $$ \begin{aligned} &u(0, t)=0, \quad u(1, t)=0, t>0 \\ &u(x, 0)=0,\left.\frac{\partial u}{\partial t}\right|_{t=0}=0,0

In Problems \(1-10\), solve Laplace's equation (1) for a rectangular plate subject to the given boundary conditions. \(u(0, y)=0, u(1, y)=0\) \(u(x, 0)=100, u(x, 1)=200\)