You can't solve a single differential equation for two different functions any more than you can solve a single algebraic equation for two different numbers.
I would say that you may be able to solve this using the method of Frobenius, which ultimately means that you break it down into a root series problem. Although, I believe the person above me was correct in saying that you cannot solve this using traditional methods since you have three different variables.
hmm I wonder if its possibly to rework this into a partial differential equation for x that is a function of u and y, while not giving you a nice litle functionfor u and y it will give you a general idea of the solution.
#5
SGT
This equation seems to represent a dynamic system, where u is the input and y the output.
You can use Laplace transforms to get a transfer function from u to y.
To solve for y you must know the function u(x) and the initial conditions y(0) and y'(0).