A system of PDEs

Question

Consider the following system for $u(x,y)$, $v(x,y)$.

$$2x^2yu_x + 5xy^2u_y + 2x^2y^2v_y + 5xyu + x = yu_y - x^2v_x + u - 2xv = 0 $$

Prove that it is equivalent to a second order semilinear PDE.

Any hints appreciated!

Answer 1

First derive this one wrt $x$

$2x^2yu_x + 5xy^2u_y + 2x^2y^2v_y + 5xyu + x =0$

$$4xyu_x+2x^2yu_{xx}+5y^2u_y+5xy^2u_{yx}+\color{red}{4xy^2v_y+2x^2y^2v_{yx}}+5yu+5xyu_x+1=0\tag 1$$

Now derive the second wrt $y$

$yu_y - x^2v_x + u - 2xv = 0$

$u_y+yu_{yy}-x^2v_{xy}+u_y-2xv_y=0$, rewrite it as

$$2u_y+yu_{yy}=x^2v_{xy}+2xv_y\tag 2$$

Note that in $(1)$ we have almost the same expression with the partial derivatives of $v$ as in $(2)$. We can substitute them by first multiplying $(2)$ by $2y^2$ (we consider that $v_{xy}=v_{yx}$, of course)

$4y^2u_y+2y^3u_{yy}=2y^2x^2v_{xy}+4y^2xv_y$

$4xyu_x+2x^2yu_{xx}+5y^2u_y+5xy^2u_{yx}+\color{red}{4y^2u_y+2y^3u_{yy}}+5yu+5xyu_x+1=0$

Well, it is a second order semilinear pde in $u$ only.