
## 1. Problems to Chapter 1

Problem 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. 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. 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. 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. 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*}