As you see I added a 10 volt voltage source to represent the initial charge on the capacitor. Then I drew the equivalent laplace circuit representing the impedance of the capacitor and inductor in terms of S.

The object here is to find the current I(t) in the mesh.

So I write KVL.

Now to guess at the solution, I check the poles by setting the denominator = 0.

I get the following. S1= -200 plus 400 j S2= -200 minus 400 j

So the solution should be of the form A e^-200t sin(400t) + B e^-200t cos(400t) or so I remember from differential equations class.

So now I need to do partial fractions. Okay in my book I have this equation.

My question is, how do I use that equation or ANYTHING ELSE to get the inverse laplace of my expression for i(s) above?

The book says the answer is

i(t)= -1/40 e^-200t *sin(400t)

So I know for sure that my expression for i(s) is correct. How do I get i(t) though. Please help. I am having real trouble with complex conjugate pairs in laplace circuits.