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?
