Not that hard. As @William Rose sugggests, the mass matrix solves it, but you need to look carefully at the problem.
J is given as a function of the other derivatives. But J appears only in the other equations in a simple form. So, substitute into equations 1 and 2 for J. Now, move all the derivative terms to the left hand side. Collect terms. I won't do the work for you, because the notation is nasty to write. (The symbolic TB should help in this if you want, but it is just basic algebra.) And anyway, you don't give the constants, so I cannot show the solution as an example.
In the end, you will have a mass matrix for a TWO equation system, in TWO variables, C and phi. So the mass matrix will be 2x2, a function of the variables, but that is not a problem. You have two initial values at x==0, so ODE45 should have no problem, unless there is a singularity in the mass matrix. I'd worry about that when there is a reason to worry. A singularity should arise only if A == 2*C(x), so you might watch for that case. Regardless, that was going to kill you in any event.
Once you have solved for C(x) and phi(x), then you can recover J(x).
As I said, not difficult.