Linear ODE Solutions Without Initial Conditions and the Arbitrary Constant C

[jdstokes]

Suppose I already have a solution $u$ to a first order ODE.

If I try to solve this ODE without initial conditions and I get another solution $w$, then it can be regarded as a function of an arbitrary constant: $w=w(C)$.

Is it true to say that $u = w(C)$ for some C? If so, how do I find such a C?

 

[Dick]

That's a little vague. You use the boundary conditions to find C. In general, an ODE doesn't even have a unique solution unless you make some assumptions about the form of the ODE. Can you be more concrete?