91Ó°ÊÓ

In Problems 17-26, classify the given partial differential equation as hyperbolic, parabolic, or elliptic. $$ \frac{\partial^{2} u}{\partial x^{2}}+6 \frac{\partial^{2} u}{\partial x \partial y}+9 \frac{\partial^{2} u}{\partial y^{2}}=0 $$

Classify the given partial differential equation as hyperbolic, parabolic, or elliptic.\(\frac{\partial^{2} u}{\partial x^{2}}+6 \frac{\partial^{2} u}{\partial x \partial y}+9 \frac{\partial^{2} u}{\partial y^{2}}=0\)

Consider the boundary-value problem $$ \begin{aligned} &k \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t}, 00 \\ &u(0, t)=u_{0}, u(L, t)=u_{1} \\ &u(x, 0)=f(x) \end{aligned} $$ that is a model for the temperature \(u\) in a rod of length \(L\). If \(u_{0}\) and \(u_{1}\) are different nonzero constants, what would you intuitively expect the temperature to be at the center of the rod after a very long period of time? Prove your assertion.

In Problems 17-26, classify the given partial differential equation as hyperbolic, parabolic, or elliptic. $$ \frac{\partial^{2} u}{\partial x^{2}}-\frac{\partial^{2} u}{\partial x \partial y}-3 \frac{\partial^{2} u}{\partial y^{2}}=0 $$

Classify the given partial differential equation as hyperbolic, parabolic, or elliptic.\(\frac{\partial^{2} u}{\partial x^{2}}-\frac{\partial^{2} u}{\partial x \partial y}-3 \frac{\partial^{2} u}{\partial y^{2}}=0\)

Classify the given partial differential equation as hyperbolic, parabolic, or elliptic.\(\frac{\partial^{2} u}{\partial x^{2}}=9 \frac{\partial^{2} u}{\partial x \partial y}\)

In Problems 17-26, classify the given partial differential equation as hyperbolic, parabolic, or elliptic. $$ \frac{\partial^{2} u}{\partial x^{2}}=9 \frac{\partial^{2} u}{\partial x \partial y} $$

Solve the Neumann problem for a rectangle: $$ \begin{aligned} &\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0,0

Classify the given partial differential equation as hyperbolic, parabolic, or elliptic.\(\frac{\partial^{2} u}{\partial x^{2}}+2 \frac{\partial^{2} u}{\partial x \partial y}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial u}{\partial x}-6 \frac{\partial u}{\partial y}=0\)

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