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Problem 1.
Solve IVP \begin{equation} \left\{\begin{aligned} &u_{tt}-c^2u_{xx}= f(x,t);\\ &u|_{t=0}=g(x),\\ &u_t|_{t=0}=h(x) \end{aligned} \right. \end{equation} with \begin{align} &f(x,t)=\sin(\alpha x), && g(x)=0, && h(x)=0;\\[4pt] &f(x,t)=\sin(\alpha x)\sin (\beta t),&& g(x)=0,&& h(x)=0;\\[4pt] &f(x,t)= f(x), && g(x)=0, && h(x)=0; \label{1a}\\[4pt] &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 $f(x)=F'''(x)$.
Hint. You may apply D'Alembert formula (2.4.11).
Alternatively, one can try to solve first the equation (without initial conditions). If the right-hand expression either does not depend on $t$ we can try $v=v(x)$; or one can try $v(x,t)= \phi(x)\psi(t)$. Then look at $u(x,t)=v(x,t)+w(x,t)$ where $w$ satisfies $w_{tt}-w_{xx}=0$; what initial conditions should satisfy $w(x,t)$?
Problem 2.
Problem 3. Find formula for solution of the Goursat problem \begin{equation} \left\{\begin{aligned} &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{aligned} \right. \end{equation} provided $g(0)=h(0)$.
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 4. Find the general solutions of the following equations: \begin{align} &u_{xy}=u_{x}u_{y} u^{-1};\\[4pt] &u_{xy}=u_{x}u_{y};\\[4pt] &u_{xy}=\frac{u_{x}u_{y} u}{u^2+1}. \end{align}