$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ ##[1. Problems to Chapter 1](id:sect-1.P) > 1. [Problem 1](#problem-1.P.1) > 2. [Problem 2](#problem-1.P.2) > 3. [Problem 3](#problem-1.P.3) > 4. [Problem 4](#problem-1.P.4) > 5. [Problem 5](#problem-1.P.5) **[Problem 1.](id:problem-1.P.1)** Consider first order equations and determine if they are linear homogeneous, linear inhomogeneous or non-linear ($u$ is an unknown function): \begin{gather\*} u\_t+xu\_x= 0,\\\\ u\_t+uu\_x= 0,\\\\ u\_t+xu\_x- u=0,\\\\ u\_t+u u\_x+x=0,\\\\ u\_t + u\_x -u^2=0,\\\\ u\_t^2-u\_x^2-1=0,\\\\ u\_x^2+u\_y^2-1=0,\\\\ x u\_x + y u\_y+ zu\_z=0,\\\\ u\_x^2 + u\_y^2+ u\_z^2-1=0,\\\\ u\_t + u\_x^2+u\_y^2=0. \end{gather\*} For non-linear equations determine if they are *quasilinear* (quasilinear= linear with respect to first-order derivatives $(u\_x,u\_y)$, but not to derivatives and function itself $(u\_x,u\_y,u)$. **[Problem 2.](id:problem-1.P.2)** Consider equations and determine their order; determine if they are linear homogeneous, linear inhomogeneous or non-linear ($u$ is an unknown function): \begin{gather\*} &u\_t+ (1+x^2)u\_{xx}=0,\\\\ &u\_t- (1+u^2)u\_{xx}=0,\\\\ &u\_t +u\_{xxx}=0,\\\\ &u\_t +uu\_x+u\_{xxx}=0,\\\\ &u\_{tt}+u\_{xxxx}=0,\\\\ &u\_{tt}+u\_{xxxx}+u=0,\\\\ &u\_{tt}+u\_{xxxx}+\sin(x)=0,\\\\ &u\_{tt}+u\_{xxxx}+\sin(x)\sin(u)=0. \end{gather\*} **[Problem 3.](id:problem-1.P.3)** Find the general solutions to the following equations \begin{align\*} u\_{xy}&=0,\\\\ u\_{xy}&= 2u\_x,\\\\ u\_{xy}&=e^{x+y},\\\\ u\_{xy}&= 2u\_x+e^{x+y}. \end{align\*} *Hint:* Introduce $v=u\_x$ and find it first. **[Problem 4.](id:problem-1.P.4)** Find the general solutions to the following equations \begin{align\*} u u\_{xy}&=u\_xu\_y,\\\\ u u\_{xy}&= 2u\_xu\_y,\\\\ u\_{xy}&=u\_x u\_y \end{align\*} *Hint:* Divide two first equations by $uu\_x$ and observe that both the right and left-hand expressions are derivative with respect to $y$ of $\ln (u\_x)$ and $\ln (u)$ respectively. Divide the last equation by $u\_x$. **[Problem 5.](id:problem-1.P.5)** Find the general solutions to the following equations \begin{align\*} u\_{xxyy}&=0, \\\\ u\_{xyz}&= 0,\\\\ u\_{xxyy}&=\sin(x)\sin(y),\\\\ u\_{xyz}&= \sin(x)\sin(y)\sin(z),\\\\ u\_{xyz}&= \sin(x)+\sin(y)+\sin(z). \end{align\*} ________ [$\Leftarrow$](./S1.4.html)  [$\Uparrow$](../contents.html)  [$\Rightarrow$](../Chapter2/S2.1.html)