# RLC differential equation – Battery exchange mathematics

I'm trying to find the current in $$t = frac { pi} {4}$$

The RLC series circuit has a voltage source given by $$E (t) = sin (100t)$$ with a 2 F capacitor, a 0.02 ohm resistor and an 0.001H inductor. If the initial current and the initial charge in the capacitor are both zero,
determine the current in the circuit to $$t = frac { pi} {4}$$

Based on the given information I have the following equation:

$$0.001 frac {d ^ 2I} {dt ^ 2} +0.02 frac {dI} {dt} + 500I = 100000cos (100t)$$

so the associated homogeneous equation is: $$r ^ 2 + 20r + 500 = (r + 10) ^ 2 + 20 ^ 2 = 0$$

whose roots are $$-10 20$$. Therefore, the solution to the homogeneous equation is:

$$I_h = C_1e ^ {- 10t} cos (20t) + C_2e ^ {- 10t} sin (20t)$$

To find a particular solution, I use the indeterminate coefficient method.

$$I_p = -10.080 cos (100t) +2.122 sin (100t).$$

And after finding $$C_1$$ Y $$C_2$$ I have:

$$I (t) = e ^ {- 10t} (10,080 cos (100t) -5,570 sin (20t)) – 10,080cos (20t) + 2,122sin (20t)$$

But $$I ( frac { pi} {4}) = 0.080111$$, and the answer that the professor gave was 10.076. I am doing something wrong?