# Laplace transform IVP

1. Feb 14, 2010

### bitrex

1. The problem statement, all variables and given/known data
I'm trying to use the Laplace transform to find the charging equation of a capacitor with an initial voltage already on the capacitor.

2. Relevant equations

$$V_i = RC\frac{dVc}{dt} + V_c$$

The initial condition of Vc will be $$\lambda$$.

3. The attempt at a solution

$$V_i(s) = sRC*V_c(s) + V_c(s) - \lambda$$

Vi will be a step function of magnitude V, so

$$\frac{V_i}{s} + \lambda = (sRC + 1)V_c(s)$$

$$V_c(s) = \frac{V}{s(sRC+1)} + \frac{\lambda}{sRC+1}$$

So using a table of Laplace transforms I get:

$$V_c(t) = V_i(1-e^{\frac{-t}{RC}}) + \frac{\lambda}{RC}(e^{\frac{-t}{RC}})$$

The problem is that this doesn't seem to agree with the formula I've seen, it's close but lambda shouldn't be divided by RC. I'm wondering how I went wrong? Thanks for any advice.

2. Feb 14, 2010

### vela

Staff Emeritus
Your mistake when you took the Laplace transform of dVc/dt:

$$L[RC V']=RC L[V']=RC(sV(s)-\lambda)$$

3. Feb 14, 2010

### bitrex

Whoops! Thanks for the catch...