Problems to Section 2.4

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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;\\ &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. 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 3. Find the general solutions of the following equations: \begin{equation} \left\{\begin{aligned} &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{aligned} \right. \end{equation}

Problem 4.

  1. Find solution $u(x,t)$ to \begin{equation} \left\{\begin{aligned} &u_{tt}-u_{xx}= (x^2-1)e^{-\frac{x^2}{2}},\\ &u|_{t=0}=-e^{-\frac{x^2}{2}}, \quad u_t|_{t=0}=0. \end{aligned} \right. \end{equation}
  2. Find $\lim _{t\to +\infty} u(x,t)$.

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