# Electrodynamics: Solve the differential equation

1. Nov 20, 2008

### Niles

1. The problem statement, all variables and given/known data
Hi all.

Please take a look at the attached circuit, where the current is direct (i.e. DC).

I have found the following differential equation using Kirchoff's laws:

$$V=\frac{R_1+R_2}{R_2C}Q+R_1\frac{dQ}{dt}.$$

I wish to solve this equation, and thus to find Q(t).

3. The attempt at a solution

First I differentiate with respect to t to obtain:

$$0=\frac{R_1+R_2}{R_2C}\frac{dQ}{dt}+R_1\frac{d^2Q}{dt^2}.$$

Solving this gives me the following:

$$Q(t)=A+B\exp\left( {\frac{R_2+R_1}{R_2R_1C}t} \right),$$

where I have used the fact that Q(t=0)=0 to find that B=-A. But how do I find A?

Sincerely,
Niles.

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Last edited: Nov 21, 2008
2. Nov 20, 2008

### Feldoh

Obtain an expression for the current, then find the charge.

Just use the fact the i = dq/dt and di/dt = d^2q/dt^2 and use separation of variables.

3. Nov 20, 2008

### naresh

You need two initial conditions to find the two constants. One is Q(0), the other one? What is the initial current?

4. Nov 21, 2008

### Niles

I am not told what I(0) is, but I guess it is zero. But how will this help me?

And if I solve for I instead of Q, then how can I find the particular solution?

5. Nov 21, 2008

### Niles

Ok, I solved it. I just have to find the final charge on the capacitor, and that is the particular solution.