# Solving RC filter by differential equation

I'm trying to solve the differential equation of a simple RC filter. What i got so far is:

$$CRV'+V=V_0\sin(\omega t)$$

Solving i get:

$$V == \frac{V_0 \sin(\omega t)}{1 + R^2C^2 \omega^2} - \frac{RCV_0 \omega \cos(\omega t)}{1 + R^2C^2 \omega^2} +c_1e^{-\frac{t}{RC}}$$

Can someone please help me understand how to interpret this with the usual concepts of reactance and impedance?

simulate this circuit – Schematic created using CircuitLab

• It is probably a lot easier to understand using complex numbers. Complex numbers make solving trig equation a lot easier. en.wikipedia.org/wiki/RC_circuit – SamFisher83 Sep 15 '15 at 5:29

For steady state sinusoidal response, the exponential term goes to zero and you are left with sine and cosine terms that can be expressed as amplitude and phase angle. The same result can be obtained by using Laplace analysis (replace C with the reactance 1/sC , followed by $s\rightarrow j\omega$) to find the steady state response.