Electric Circuit Lapalace Transform Question

*Laplace Transform.

I need help with solving this equation, please.

In an RLC circuit, L =1H, C = 1/5F, R =2Ohm and f(t) = 2 sin t. Write down the differential equation that describes v(t), the voltage across capacitor, and solve it with initial conditions v(0) = 0 and dv(0)/dt = 3. Ans = - cos t + 2 sin t + e^(-t) (cos 2t + sin 2t)

So far I've managed to obtain the equation of V(s) = [F(s) + 3LC] / [LC s^2 +RCs + 1 ]
RC = 2/5
LC = 1/5
F(s) = 1 /(s^2 + 1)

V(s) = [(1/(s^2 + 1)) + 3/5] / [(1/5)s^2 + (2/5)s + 1]

I tried to solve it by doing partial equation but ended up with wrong answers.

You've posted what you get when you take the laplace transform, but what is your original differential equation?

My original differential equation is:

LC v(t)" + RC v(t)' + v(t) = f (t)

What I've managed so far:
1. Breaking V(s) into parts A and B, part A would then branch to parts C and D
http://i53.tinypic.com/2cd7tyd.jpg

2. Solving part C
http://i54.tinypic.com/28rm903.jpg

3. Solving parts D and B from earlier (continued on next image)
http://i54.tinypic.com/24cxxf6.jpg

4. Solving the rest of part B and putting all the parts together.
http://i55.tinypic.com/287kbw4.jpg

If my last answer at the bottom of the last image is multiplied by 2, I would be able to obtain the given answer EXCEPT for component (5/4)e^(-t) sin 2t. I'm not sure if I have inversed the transformation with a mistake somewhere, please do check.