Question: How to calculate: Charging Capacitor Arbitrary waveform

Charging a capacitor with a DC is well known and interaction with a sinusoidal signal is known to perfection.

The mathematical solutions behind an arbitrary waveform V1(t) is hard or impossible? to find .

To simplify, the arbitrary wave can be expressed with a 2-d degree polynom. This gives a fairly simple

expression with an integral. The problem however is that V2(t) - the capacitor voltage - appears on both sides and maple seems unable to resolv this into a symbolic expression.

If this can be solved it would be great ! The solution should be of the symbolic form V2(a,b,c,V2start,t1,t2,RC) and give the answer what voltage the capacitor V2 would reach arter charging between t1 and t2  with a V1(t) that is changing during charge.