@J4James This de actually looks somewhat similar to the equation of a plane electro-magnetic wave in a circular waveguide.
As others already said, _C1 (the coefficient of BesselY(0,0)) should be 0 else you end up with an infinity at r=0. Of course this holds only if indeed you need defined behaviour of the solution at r=0... your choice for the initial conditions indicates that you do.
For such equations, you find _C2 (the coefficient of BesselJ(0,0)) from the value of the solution on the axis (r=0); here we have w(0)=_C2. Note that your solution should not be w(0)=0 as that is the trivial solution (w(r)=0 for any r).
For given values of the other parameters this defines your solution. As you found out, diff(w(r),r) is not useful here as it is 0 at r=0 (once you got rid of the BesselY term). Note that this is a mathematically valid solution for sure, even if it may not be the one you are looking for.
You may also want to know r for w(r)=0. (In the wave equation this would be the boundary condition at the radius of the waveguide). This is tricky here as you have the other terms besides the BesselJ term and Maple cannot solve this directly. fsolve can possibly be coerced to give you a solution but would need values for the various parameters. You can also try a series expansion and get an estimate for the 0 crossing of w(r) (I'd try 2nd order (3 terms in series()) first).
All of which begs the question: What kind of system is your de actually describing? If this describes a physical system you can probably glean the correct initial conditions from that.