Consider the following equation: $$x\frac{\partial}{\partial x}f(x,y) + A f(x,y) + B\frac{\partial}{\partial y}f(x,y) + C\frac{\partial^2}{\partial y^2}f(x,y)+D\frac{\partial^3}{\partial y^3}f(x,y)=0$$ Where $A,B,C,D$ are some chosen real numbers. Let $a_k(x) = \frac{\partial^k}{\partial y^k}f(x,0)$, for $k=0,1,2,3$
Solve this equation for $f:\mathbb{R}\times \mathbb{R}\rightarrow\mathbb{R}$.
I was trying to apply Laplace transform for function $h_{x}(y) = f(x,y)$, but this gave me no result. However i would like to show you my attempt.
Consider function $F(x,s) = \int_{0}^{\infty}f(x,y)e^{-sy}dy$. Then i may rewrite the equation in following way: $$x\frac{\partial}{\partial x}F(x,s) + A F(x,s) + BsF(x,s)+Cs^2F(x,s)+Ds^3F(x,s)=H(s)$$ where $H(x,s)$ is some function that consists $a_k(x)$ multiplied by some powers of $s$.
I have noticed that this is ordinary differential equation.
Next, after solving this new equation, i would to use inverse Laplace transform.
But i don't know how to solve even this new equation.
Thank you for help.