What is this motivation for changing variables to standardize a pde? In a question I am looking at it says -
Find a change of variables $u \rightarrow v$ of the form $u = ve^{ax + by}$ that would transform the PDE
$$\frac{\delta^2u}{\delta x^2} + \frac{\delta^2u}{\delta y^2} + \frac{\delta u}{\delta x} + 2\frac{\delta u}{\delta y} + 3u = 0$$
into
$$\frac{\delta^2v}{\delta x^2} + \frac{\delta^2v}{\delta y^2} + Av = 0$$
where $A$ is a constant.
What is the motivation for the change of variables being of the form $u = ve^{ax + by}$ that lets us get the pde in 'nicer' form?

The motivation comes from solving ODE with the method of integration factors. Suppose you're to solve $y'+2y=f(x)$. A natural approach is to let $z=e^{ax}y$ and compute $z'=e^{ax}(y'+ay)$. In order to match the original ODE, we let $a=2$. Then $z'=e^{2x}f(x)$, which we can integrate to solve $z$, and hence $y=e^{-2x}z$.
