Voltage across capacitor in charging RC circuit with constant current source
simulate this circuit &ndash; Schematic created using CircuitLab

As @TimWescot said, I solved it by transforming the circuit to it's Thevenin Equivalent which has a voltage source that allows for integration using separation of variables.

$$ \frac{V_c - V_s}{R} + C\frac{dV}{dt} = 0$$

Excuse the values since I'm only working with variables and not concerned about the actual values.