Chapter 13

In Problems 17-26, classify the given partial differential equation as hyperbolic, parabolic, or elliptic. $$ \frac{\partial^{2} u}{\partial x^{2}}+2 \frac{\partial^{2} u}{\partial x \partial y}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial u}{\partial x}-6 \frac{\partial u}{\partial y}=0 $$

Classify the given partial differential equation as hyperbolic, parabolic, or elliptic.\(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=u\)

In Problems 17-26, classify the given partial differential equation as hyperbolic, parabolic, or elliptic. $$ \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=u $$

Classify the given partial differential equation as hyperbolic, parabolic, or elliptic.\(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=u\)

In Problems 17-26, classify the given partial differential equation as hyperbolic, parabolic, or elliptic. $$ a^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}} $$

A model for an infinitely long string that is initially held at the three points \((-1,0),(1,0)\), and \((0,1)\) and then simultaneously released at all three points at time \(t=0\) is given by (13) with $$ f(x)=\left\\{\begin{array}{ll} 1-|x|, & |x| \leq 1 \\ 0, & |x|>1 \end{array} \text { and } g(x)=0 .\right. $$ (a) Plot the initial position of the string on the interval \([-6,6]\). (b) Use a CAS to plot d'Alembert's solution \((14)\) on \([-6,6]\) for \(t=0.2 k, k=0,1,2, \ldots, 25\). Assume that \(a=1\). (c) Use the animation feature of your computer algebra system to make a movie of the solution. Describe the motion of the string over time.

Classify the given partial differential equation as hyperbolic, parabolic, or elliptic.\(a^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}\)

In Problems 17-26, classify the given partial differential equation as hyperbolic, parabolic, or elliptic. $$ k \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t}, k>0 $$

Classify the given partial differential equation as hyperbolic, parabolic, or elliptic.\(k \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t}, k>0\)

Show that the given partial differential equation possesses the indicated product solution.\(k\left(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}\right)=\frac{\partial u}{\partial t}\) \(u=e^{-k \alpha^{2} t}\left(c_{1} J_{0}(\alpha r)+c_{2} Y_{0}(\alpha r)\right)\)

In Problems 27 and 28 , show that the given partial differential equation possesses the indicated product solution. $$ \begin{aligned} &\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}=0 \\ &u=\left(c_{1} \cos \alpha \theta+c_{2} \sin \alpha \theta\right)\left(c_{3} r^{\alpha}+c_{4} r^{-\alpha}\right) \end{aligned} $$

Discuss whether product solutions \(u=X(x) Y(y)\) can be found for the given partial differential equation. [Hint: Use the superposition principle.]\(\frac{\partial^{2} u}{\partial x^{2}}-u=0\)

Discuss whether product solutions \(u=X(x) Y(y)\) can be found for the given partial differential equation. [Hint: Use the superposition principle.]\(\frac{\partial^{2} u}{\partial x \partial y}+\frac{\partial u}{\partial x}=0\)