Proving RC and RL circuits as integrator and differentiator using differential equations
Just as linear resistors are defined by Ohm's Law, there are defining equations for linear capacitors and inductors. For a capacitor

$$Q = CV$$

or, in terms of a derivative

$$I = C\frac{\rm{d}V}{\rm{d}t}$$

For an inductor

$$\frac{\rm{d}I}{\rm{d}t}=\frac{V}{L}$$.

Combine these with Kirchoff's Laws and you will have differential equations describing your circuit.