1.P. Problems to Chapter 1

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Problems to Chapter 1

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.


$\Leftarrow$  $\Uparrow$  $\Rightarrow$