Something's gone amiss. The units in the first term do not work out to current (amps). And I'd expect a constant offset for the second term, not a decaying exponential. But it's really close to what I'd expect.l2(t) = VsR1/(R1 + R2)(1 - e-t/τ) + Vs/R2e-t/τ
And I'd expect a constant offset for the second term, not a decaying exponential.
That is the "starting from the basics" approach, yes.
One writes the circuit equations using the differential or integral forms for the capacitor voltage or current, then solve the resulting differential equation. So for example, for the simple case of a charged capacitor discharging through a resistor, writing KCL:How does one go about finding the differential equation?