TITLE: Stuck on this linear and homogenous PDE
QUESTION [5 upvotes]: $$\frac{1}{2}\frac{\partial^2w}{\partial y^2}+\sqrt{8x^2+25}\frac{\partial w}{\partial y}+4x^2w=0$$
"Find the general solution of the PDE in $w(x,y)$. State clearly any arbitrary functions that appear in general solutions."
Progression:
So far I tried using the method of separation, letting
$$w(x,y) = X(x)Y(y)$$
after subbing the partial derivatives into the original PDE, I divided the equation by $X(x)Y(y)$ and I got the following:
$$\frac{1}{2}\frac{Y''(y)}{Y(y)} + \sqrt{8x^2+25}\frac{Y'(y)}{Y(y)}+4x^2=0 $$
so usually I would let $X$ on one side and $Y$ on the other side of the equation that = a constant $\gamma$ and decompose into 2 separate ODEs but in this case, I can't find a way to separate $X$ and $Y$.
Sorry for the previous post I will delete it. This is not an assignment just a past year's exam question I was practicing. Thanks in advance for any help!

REPLY [3 votes]: Hint: Consider the simpler linear DE $\frac{\partial w}{\partial y}+x w=0$. This is equivalent to $$\frac{1}{w}\frac{\partial w}{\partial y}=\frac{\partial}{\partial y}\ln w=-x,$$
which we may immediately integrate w/r/t $y$ to obtain
$$\ln w(x,y)=-x y+C(x)\implies w(x,y)=A(x)e^{-x y}.$$
Note that this is basically equivalent to solving the linear ODE $\frac{dw}{dy}+xw=0$, treating $x$ as a constant, and then understanding that the coefficient of the general solution will depend on $x$ as well. Can you apply this insight to your DE?