Electrodynamics: Solve the differential equation

[Niles]

Homework Statement

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).

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?

Thanks in advance.Niles.

 

[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.

 

[naresh]

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

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

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

 

[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?

 

[Niles]

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