# Using the method of characteristics with an extra term

How do you apply the method of characteristics to get the solution to the following PDE:

$$xU_x-yU_y-xU=x^2y$$

$\dfrac{dx}{dt}=x$ , letting $x(0)=1$ , we have $x=e^t$
$\dfrac{dy}{dt}=-y$ , letting $y(0)=y_0$ , we have $y=y_0e^{-t}=\dfrac{y_0}{x}$
$\dfrac{dU}{dt}=xU+x^2y=e^tU+y_0e^t$ , we have $U(x,y)=f(y_0)e^{e^t}-y_0=f(xy)e^x-xy=F(xy)e^x$
• I am not sure about this: $f(xy)e^x-xy=F(xy)e^x$. Considering it true I get $(f(xy)-F(xy))=xye^{-x}$ It seems to me not possible for a single argument function. – Rafa Budría Jun 13 '18 at 12:22