Linear second order PDE with Differential Operator

Question

I have a question about how to interpret this equation:

D(xDy)-3Dy+2/x*y=1

I think it is a second order linear PDE and have simplified it to: y''x+y'(x'-3)+2/x*y=1 However, I'm not sure whether y and x depends on a third variable (like t). Moreover, I think it might be possible to factorize the equation or maybe tranform it into a ODE. However, I have only learn to solve ODE's yet.

Guidance is strongly appreciated!

Thanks in advance, Casper

Answer 1

Without other informations ( and if I well interpret) it seems the equation: $$ (xy')' -3y'+\frac{2}{xy}=1 $$ because $D$ is usually the symbol for the derivative operator. So, with $y=y(x)$ becomes: $$ y'+xy''-3y'+\frac{2}{xy}=1 $$ that is a non-linear second order ODE.