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{ \large \textbf{\textsc{Math 5587 -- Homework 2 (Due Thursday Sept 15)}}}
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\begin{enumerate}
\item  For each of the following equations, state the order and whether it is nonlinear, linear inhomogeneous, or linear homogeneous. 
\begin{enumerate}
\item $u_t - u_{xx} + x = 0$.
\item $(x+y)u_{xy} + 2xu_y = x^2$.
\item $u_{xx} = e^u$.
\item $u_{txy} - u_{xx}u_{yy} + u_{x} = x^3$.
\item $u_t + u_{xxxxxx} - \sqrt{1+u^2} = 0$.
\item $u_t + u_x + u_y + u/xy = 0$.
\end{enumerate}
\item Let $u_*$ be a solution of the inhomogeneous linear equation $L[u_*]=g$. Show that \emph{every} solution of $L[u]=g$ is of the form $u = u_* + v$, where $v$ is a solution of the homogeneous linear equation $L[v]=0$.
\item Find the solution to the initial value problem $u_t + u_x = 0$ satisfying $u(x,1) = x/(1+x^2)$.
\item Show that the only continuously differentiable solutions of $xu_x + yu_y=0$ on the entire plane $\R^2$ are constant functions. [Hint: Show that for any fixed $(x,y)\in \R^2$, the function $g(t) = u(xt,yt)$ is constant in $t$.]
\item \begin{enumerate}
\item Find a solution of $u_xu_y=1$ on $\R^2$ of the form $u(x,y) = f(x) + g(y)$. 
\item Find two different solutions of $u_xu_y = u$ in the domain $x\geq 0$ and $y\geq 0$ that satisfy $u(x,0) = 0$ and  $u(0,y)= 0$ for all $x \geq 0$ and $y\geq 0$. [Hint: One is trivial. For the other, look for a solution in the separable form $u(x,y) = f(x)g(y)$.]
\end{enumerate}
\item
\begin{enumerate}
\item Write down a formula for the general solution to the nonlinear PDE $u_t + u_x + u^2 = 0$.
\item Show that if the initial data $f(x)=u(x,0)$ is nonnegative and bounded $0 \leq f(x) \leq M$, then the solution exists for all $t>0$, and $u(x,t) \to 0$ as $t\to \infty$. 
\item On the other hand, if the initial data is negative at some $x$, show that the solution blows up in finite time: That is $\lim_{t \to \tau^-}u(y,t) = -\infty$ for some $\tau>0$ and $y \in \R$.
\item Find a formula for the earliest blow-up time $\tau_*>0$.

\end{enumerate}
\end{enumerate}


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