Math Help - A PDE transformation

1. A PDE transformation

Hi there,
In the notes I'm reading, they have stated the following PDE:
$u_{xx}+u_{yy}+2au_x+2bu_y+cu=d$.
Then, by defining:
$v(x,y)=e^{ax+by}u(x,y)$
The PDE becomes, in terms of v:
$v_{xx}+v_{yy}+(c-(a^2+b^2))v=e^{ax+by}d$
I don't see how they made this transformation. Should I taking partial derivatives of u as a function of v?

2. Should I taking partial derivatives of u as a function of v?
I would write u as a function of v and the exponential function there. That is,

$u(x,y)=e^{-ax-by}\,v(x,y).$

Then start taking the partial derivatives you need to assemble the ingredients of your first equation. Then plug everything in and simplify.

3. Cheers.