$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ ###Problems to Chapter 1 > * [Problem 1](#problem-1.P.1) > * [Problem 3](#problem-1.P.3) > * [Problem 5](#problem-1.P.5) **Problem 1.** Consider first order equations and determine if they are linear homogeneous, linear inhomogeneous, or nonlinear ($u$ is an unknown function); for nonlinear equations, indicate if they are also semilinear, or quasilinear. \begin{gather} u\_t+xu\_x= 0,\\\\[2pt] u\_t+uu\_x= 0,\\\\[2pt] u\_t+xu\_x- u=0,\\\\[2pt] u\_t+u u\_x+x=0,\\\\[2pt] u\_t + u\_x -u^2=0,\\\\[2pt] u\_t^2-u\_x^2-1=0,\\\\[2pt] u\_x^2+u\_y^2-1=0,\\\\[2pt] x u\_x + y u\_y+ zu\_z=0,\\\\[2pt] u\_x^2 + u\_y^2+ u\_z^2-1=0,\\\\[2pt] 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.** 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,\\\\[2pt] u\_t- (1+u^2)u\_{xx}=0,\\\\[2pt] u\_t +u\_{xxx}=0,\\\\[2pt] u\_t +uu\_x+u\_{xxx}=0,\\\\[2pt] u\_{tt}+u\_{xxxx}=0,\\\\[2pt] u\_{tt}+u\_{xxxx}+u=0,\\\\[2pt] u\_{tt}+u\_{xxxx}+\sin(x)=0,\\\\[2pt] u\_{tt}+u\_{xxxx}+\sin(x)\sin(u)=0. \end{gather} **Problem 3.** Find the general solutions to the following equations: \begin{gather} u\_{xy}=0,\\\\[2pt] u\_{xy}= 2u\_x,\\\\[2pt] u\_{xy}=e^{x+y},\\\\[2pt] u\_{xy}= 2u\_x+e^{x+y}. \end{gather} *Hint.* Introduce $v=u\_x$ and find it first. **Problem 4.** Find the general solutions to the following equations: \begin{gather} u u\_{xy}=u\_xu\_y,\\\\[2pt] u u\_{xy}= 2u\_xu\_y,\\\\[2pt] u\_{xy}=u\_x u\_y \end{gather} *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.** Find the general solutions to the following equations: \begin{gather} u\_{xxyy}=0, \\\\[2pt] u\_{xyz}= 0,\\\\[2pt] u\_{xxyy}=\sin(x)\sin(y),\\\\[2pt] u\_{xyz}= \sin(x)\sin(y)\sin(z),\\\\[2pt] u\_{xyz}= \sin(x)+\sin(y)+\sin(z). \end{gather} **Problem 6.** Find the general solutions to the following *overdetermined systems*: \begin{align} &\left\\{\begin{aligned} &u\_{xx}=0,\\\\[2pt] &u\_{y}=0; \end{aligned}\right. \\\\[2pt] &\left\\{\begin{aligned} &u\_{xx}=6xy,\\\\[2pt] &u\_{y}=x^3; \end{aligned}\right. \\\\[2pt] &\left\\{\begin{aligned} &u\_{xx}=6xy,\\\\[2pt] &u\_{y}=-x^3. \end{aligned}\right. \end{align} *Hint.* Solve one of the equations and plugging the result to another, specify an arbitrary function (or functions) in it, and write down the final answer. ________ [$\Leftarrow$](./S1.4.html)  [$\Uparrow$](../contents.html)  [$\Rightarrow$](../Chapter2/S2.1.html)