$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ ###Problems to Chapter 1 > * [Problem 1](#problem-1.P.1) > * [Problem 3](#problem-1.P.3) > * [Problem 5](#problem-1.P.5) > * [Problem 7](#problem-1.P.7) **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: [$^{1)}$](#foot-1.P.1): \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} **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 *linear homogeneous equations*: \begin{align} &Lu:=u\_{xxy}=0;\\\\[2pt] &Lu:=u\_{xxyy}=0; \\\\[2pt] &Lu:=u\_{xxxy}=0;\\\\[2pt] &Lu:=u\_{xyz}= 0;\\\\[2pt] &Lu:=u\_{xyzz}=0. \end{align} **Problem 6.** Find the general solution to the following *linear inhomogeneous equations*: \begin{equation} Lu= f \end{equation} where $L$ is defined in Problem 5 and $f=f(x,y)$ or $f=f(x,y,z)$ is given[$^{2)}$](#foot-1.P.6):. **Problem 7.** 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\_{xy}=0,\\\\[2pt] &u\_{xz}=0; \end{aligned}\right. \\\\[2pt] &\left\\{\begin{aligned} &u\_{xy}=0,\\\\[2pt] &u\_{xz}=0,\\\\[2pt] &u_{yz}=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. Often overdetermined systems do not have solutions; f.e. \begin{equation} \left\\{\begin{aligned} &u\_{x}=M(x,y),\\\\[2pt] &u\_{y}=N(x,y) \end{aligned}\right. \end{equation} has a solution iff $M_y-N_x=0$. _____ $^{1)}$ $F(x,y, u, u_x,u_y)=0$ is *non-linear* unless \begin{equation} F:= au_x+bu_y+cu-f \label{eqn-1P.A} \end{equation} with $a=a(x,y)$, $b=b(x,y)$, $c=c(x,y)$ and $f=f(x,y)$, when it is *linear homogeneous* for $f(x,y)=0$ and *linear inhomogeneous* otherwise. If \begin{equation} F:= au_x+bu_y-f \label{eqn-1P.B} \end{equation} with $a=a(x,y,u)$, $b=b(x,y,u)$ and $f=f(x,y,u)$ (so it is linear with respect to (highest order) derivatives, it is called *quasilinear*, and if in addition $a=a(x,y)$, $b=b(x,y)$, it is called *semilinear*. This definition obviously generalizes to higher dimensions and orders. $^{2)}$ $f(x,y)$ or $f(x,y,z)$ will be defined in the graded assessment; you may consider $f(x,y)=\cos(x)+\sin(y)$, $f(x,y)=\cos (x)\sin(y)$ and so on. ________ [$\Leftarrow$](./S1.4.html)  [$\Uparrow$](../contents.html)  [$\Rightarrow$](../Chapter2/S2.1.html)