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: $^{1)}$: \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); you need to provide the most precise (that means the narrow, but still correct) description: \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{gather} u_{xxy}=0,\\[2pt] u_{xxyy}=0, \\[2pt] u_{xxxy}=0,\\[2pt] u_{xyz}= 0,\\[2pt] u_{xyzz}=0,\\[2pt] u_{xxy}=\sin(x)\sin(y),\\[2pt] u_{xxy}=\sin(x)+\sin(y),\\[2pt] u_{xxyy}=\sin(x)\sin(y),\\[2pt] u_{xxyy}=\sin(x)+\sin(y),\\[2pt] u_{xxxy}=\sin(x)\sin(y),\\[2pt] u_{xxxy}=\sin(x)+\sin(y),\\[2pt] u_{xyz}= \sin(x)\sin(y)\sin(z),\\[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_{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.

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