A PDE with mixed derivative

Question

How should one go about solving an equation of the form

$$\frac{\partial^2 u}{\partial x \partial y} + x \frac{\partial u}{\partial y} = y$$

Do I need to use characteristics, or integrate first?

Answer 1

A first integration w.r.t $y$ would give $$ u_x+xu=\frac{y^2}{2}+f(x), $$ where $f(x)$ is an arbitrary function of $x$. Now we multiply the sides of the equation in $\exp(\frac{x^2}{2})$ to get $$ [\exp(\frac{x^2}{2})u]_x=\exp(\frac{x^2}{2})u_x+x\exp(\frac{x^2}{2})u=\frac{y^2}{2}\exp(\frac{x^2}{2})+f(x)\exp(\frac{x^2}{2}) $$where another integration (this time w.r.t. $x$) yields $$ \exp(\frac{x^2}{2})u=\frac{y^2}{2}[\int \exp(\frac{x^2}{2})dx+g(y)]+h(y)+\int f(x)\exp(\frac{x^2}{2})dx, $$ together giving $$ u(x,y)=\frac{y^2}{2}\exp(-\frac{x^2}{2})\int \exp(\frac{x^2}{2})dx+h_1(y)\exp(-\frac{x^2}{2})+h_2(x) $$