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<title>Problems to Section 2.4</title>

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###[Problems](id:sect-2.4.P)

> 1. [Problem 1](#problem-2.4.P.1)
> 2. [Problem 2](#problem-2.4.P.2)
> 3. [Problem 3](#problem-2.4.P.3)


**[Problem 1.](id:problem-2.4.P.1)**

Solve IVP
\begin{align}
&u\_{tt}-c^2u\_{xx}= f(x,t);\\\\
&u|\_{t=0}=g(x),\\\\
&u\_t|\_{t=0}=h(x)
\end{align}
with
\begin{align}
&f(x,t)=\sin(\alpha x), && g(x)=0, && h(x)=0;\\\\
&f(x,t)=\sin(\alpha x)\sin (\beta t), && g(x)=0; && h(x)=0,\\\\
&f(x,t)= f(x),  && g(x)=0, && h(x)=0; \label{1a}\\\\
&f(x,t)= f(x)t,  && g(x)=0, && h(x)=0, \label{1b}
\end{align}
in the  case (\ref{1a}) assume that $f(x)=F''(x)$ and in the case (\ref{1b}) assume that $f(x)=F'''(x)$.

**[Problem 2.](id:problem-2.4.P.2)**
Find formula for solution of the *Goursat problem*
\begin{align}
&u\_{tt} - c^2 u\_{xx}=f(x,t), && x \> c|t|, \\\\
&u|\_{x=-ct}=g(t),  && t\<0, \\\\
&u|\_{x=ct}=h(t),   &&t \> 0
\end{align}
as $g(0)=h(0)$.

*Hint.* Contribution of the right-hand expression will be
\begin{equation}
-\frac{1}{4c^2}\iint _{R(x,t)} f(x',t')\,dx'dt'
\end{equation}
with $R(x,t)=\\{ (x',t'):\, 0\< x'-ct' \< x-ct,\, 0\< x'+ct' \< x+ct\\}$.

**[Problem 3.](id:problem-2.4.P.3)**
Find the general solutions of the following equations:
\begin{align}
&u\_{xy}=u\_{x}u\_{y} u^{-1};\\\\
&u\_{xy}=u\_{x}u\_{y};\\\\
&u\_{xy}=\frac{u\_{x}u\_{y} u}{u^2+1};
\end{align}

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