Can anybody help me with this question?

Question

1. An electric circuit contains a resistance R and a capacitor C in series, and a

battery supplying a time-varying electromotive force V(t). The charge q on

the capacitor therefore obeys the equation

R(dq/dt) + q/C = V(t)

Assume that initially there is no charge on the capacitor, and given that V (t) =

V0 sin(t)

(a) Find the Integrating Factor for the above equation.

(b) Solve the differential equation.

Hint: You might find using q(0) = 0 useful.

Answer 1

I don't see an easy way to solve this using an integrating factor.

If the right hand side is written as V0 sin(ωt) [ take ω = 1 at the end ].

the conventional approach is to note that the solution to the homogeneous equation is

q(t) = Q0 exp( - t /RC) for some Q0.

The next step is to look for a particular solution to the full equation having the form

q = A sin( ωt - p) . After a lot of algebra one finds that

tan(p) = ωRC

sin(p) = 1 / ( 1 + (ωRC)^2 )^.5 and

A = [V0 C / (ωRC) ] / ( 1 + (ωRC)^2) ^.5

The full solution is q = Q0 exp(-RC/t) + A sin(ωt - p)

Setting this equal to zero for t = 0 gives:

Q0 = [V0 C /(ωRC) ] / [ 1+ 1/(ωRC)^2 ]