this is my differential equation course. I got back into school after a couple of years. This is just the start of this course and I am having difficulties in one of these practice problems.
This is the question:
Find the general form of a function $U(x,y)$ solving the PDE $u_{xx} + yu_x = 0$ (1)
My attempt:
we can write (1) as
(d/dx)(du/dx) + y(du/dx) = 0 $\rightarrow$ (du/dx)[d/dx + y] = 0.
Now from here I am getting confused. Any help or hints would be greatly appreciated. Thank you.
Fix $y$ and name $f(x)=u(x,y)$. Then $f$ satisfies the ODE: $$ f''+yf'=0 $$ The general solution for this ODE is $$ f(x)=a+be^{-y\,x} $$ where $a$ and $b$ are constants.
Therefore the general form for $u(x,y)$ is $$ u(x,y)=a(y)+b(y)e^{-y\,x} $$ where $a$ and $b$ are arbitrary functions.