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\title{MTH 352/652:
Homework \#1}
\vspace{-1in}
\date{Due Date: January 26, 2024}
\maketitle

\section{Problems for Everyone}
\begin{enumerate}
\item Sign up for Piazza. I will check the roster for your name.
\item Determine which of the following operators are linear. If an operator is linear, prove it. If an operator is nonlinear find a counterexample, i.e., two functions $u,v$ for which $L[u+v]\neq L[u]+L[v]$ or a function $u$ for which $L[cu]\neq cL[u]$.
\begin{enumerate}
\item $L[u]=u_x+xu_t$
\item $L[u]=u_x+uu_t$
\item $L[u]=u_x+u_t^2$
\item $L[u]=u_x+u_t+1$
\item $L[u]=\sqrt{1+x^2}\cos(t)u_x+u_{txt}-\arctan(x/y)u$
\end{enumerate}

\item For each of the following equations, state the order and whether it is nonlinear, linear homogenous, or linear inhomogeneous. There is no need prove anything for this problem.
\begin{enumerate}
\item $u_t-u_{xx}+1=0$
\item $u_t-u_{xx}+xu=0$
\item $u_t-u_{xxt}+uu_x=0$
\item $u_{tt}-u_{xx}+x^2=0$
\item $u_{x}(1+u_x^2)^{-1/2}+u_t(1+u_t^2)^{-1/2}=0$
\item $u_x+e^{t}u_t=0$
\item $u_t+u_{xxxx}+\sqrt{1+u}=0$
\end{enumerate}
 
 \item Verify that $u(x,y)=f(x)g(y)$ is a solution to the PDE $uu_{xy}=u_xu_y$ for all pairs of differentiable functions $f$ and $g$ of one variable.
 
 \item Show that the following functions solve the PDE $u_{xx}+u_{yy}=0$:
 \begin{enumerate}
 \item $u(x,y)=e^{x}\cos(y)$
 \item $u(x,y)=1+x^2-y^2$
 \item $u(x,y)=x^3-3xy^2$
 \item $u(x,y)=\ln(x^2+y^2)$
 \item $u(x,y)=\arctan(y/x)$
 \item $\displaystyle{u(x,y)=\frac{x}{x^2+y^2}}$
 \item $u(x,y)=\sin(nx)(e^{ny}-e^{-ny})$, $n\in \mathbb{R}$

 \end{enumerate}

\item Solve the PDE $3u_t+u_{xt}=0$. \textbf{Hint:} Let $v=u_t$.
\item Find the general solution to the PDE $u_{xy}=0$ in terms of two arbitrary functions.

\item Find a function $u(t,x)$ that satisfies the PDE
\begin{equation*}
u_{xx}=0
\end{equation*}
on the domain $0<x<1$, $t>0$ subject to the boundary conditions $u(t,0)=t^2$ and $u(t,1)=1$.

\item Show that the nonlinear equation $u_t=u_x^2+u_{xx}$ can be reduced to the linear equation $w_t=w_{xx}$ by changing the variable to $w=e^{u}$.
\item Solve the following initial value problems and graph the solutions at times $t=1,2$ and $3$.
\begin{enumerate}
\item $u_t-3u_x=0$ with $u(0,x)=e^{-x^2}$
\item $2u_t+3u_x=0$ with $u(0,x)=\sin(x)$
\item $u_t+2u_x=0$ with $u(-1,x)=x/(1+x^2)$
\item $u_t+u_x+u=0$ with $u(0,x)=\arctan(x)$
\end{enumerate}

\end{enumerate}
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