$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$ ###[Problems](id:sect-2.8.P) > 1. [Problem 1](#problem-2.8.P.1) > 2. [Problem 2](#problem-2.8.P.2) > 3. [Problem 3](#problem-2.8.P.3) > 4. [Problem 4](#problem-2.8.P.4) > **[Problem 1.](id:problem-2.8.P.1)** Find the characteristics and write down the general solutions to the systems $U\_u+AU\_x=0$ with \begin{align} A&=\begin{pmatrix} 2 &3\\\\3 &2 \end{pmatrix},\\\\ A&=\begin{pmatrix} 2 &-3\\\\1 &-2 \end{pmatrix},\\\\ A&=\begin{pmatrix} 1 &-1\\\\2 &4 \end{pmatrix},\\\\ A&=\begin{pmatrix} -1 &-1\\\\2 &-4 \end{pmatrix},\\\\ A&=\begin{pmatrix} 3 &2\\\\0 & -1 \end{pmatrix},\\\\ A&=\begin{pmatrix} 3 &0\\\\ 2 & -1 \end{pmatrix} \end{align} **[Problem 2.](id:problem-2.8.P.2)** For each system from [Problem 1](#problem-2.8.P.1) in $\\{ x>0, t>0\\}$ determine which of the following IVBPs is well-posed and find solution ($U= \begin{pmatrix}u\\\\ v\end{pmatrix}$): \begin{align} &u|\_{t=0}=f(x), v|\_{t=0}=g(x); \\\\ &u|\_{t=0}=f(x), v|\_{t=0}=g(x); &&u|\_{x=o}=\phi(t),\\\\ &u|\_{t=0}=f(x), v|\_{t=0}=g(x); &&u|\_{x=o}=\phi(t), v|\_{x=0}=\psi(t) \end{align} **[Problem 3.](id:problem-2.8.3)** Find the characteristics and write down the general solutions to the systems $U\_u+AU\_x=0$ with \begin{align} A&=\begin{pmatrix} 3 &2 &1\\\\0 &2 &1\\\\ 0 &0&1 \end{pmatrix},\\\\ A&=\begin{pmatrix} 3 &2 &1\\\\0 &2 &1\\\\ 0 &0&-1 \end{pmatrix},\\\\ A&=\begin{pmatrix} 3 &2 &1\\\\0 &-2 &1\\\\ 0 &0&-1 \end{pmatrix},\\\\ A&=\begin{pmatrix} -3 &2 &1\\\\0 &-2 &1\\\\ 0 &0&-1 \end{pmatrix},\\\\ A&=\begin{pmatrix} 1 &2 &3\\\\2 &0 &3\\\\ 2 &3&0 \end{pmatrix}. \end{align} **[Problem 4.](id:problem-2.8.P.4)** For each system from [Problem 3](#problem-2.8.P.3) in $\\{ x>0, t>0\\}$ determine which of the following IVBPs is well-posed and find solution ($U= \begin{pmatrix}u\\\\ v\\\\w\end{pmatrix}$): \begin{align} &u|\_{t=0}=f(x), v|\_{t=0}=g(x), w|\_{t=0}=h(x); \\\\ &u|\_{t=0}=f(x), v|\_{t=0}=g(x), w|\_{t=0}=h(x),\\notag\\\\ &u|\_{x=0}=\phi(t);\\\\ &u|\_{t=0}=f(x), v|\_{t=0}=g(x), w|\_{t=0}=h(x),\\notag\\\\ &u|\_{x=0}=\phi(t), v|\_{x=0}=\psi(t);\\\\ &u|\_{t=0}=f(x), v|\_{t=0}=g(x), w|\_{t=0}=h(x), \\notag\\\\ &u|\_{x=0}=\phi(t), v|\_{x=0}=\psi(t), w|\_{x=0}=\chi(t). \end{align} ______________ [$\Leftarrow$](./S2.6.html)  [$\Uparrow$](../contents.html)  [$\Rightarrow$](../Chapter3/S3.1.html)