Just for completeness, I've added the solution using the Laplace transform below. The partial fractions were the toughest part, almost as tough as the double integration by parts using the integrating factor method earlier.
Megamox
**broken link removed**
Hi there,
It looks like you did a pretty good job there
To show the beauty and simplicity of the Laplace Transform methods a little more, here is another quick approach when initial conditions are all zero...
We have a source Vm*cos(wt), and we have an impedance of R in series with L driven by that source.
The Laplace Transform of the impedance Z is:
Z=R+s*L
and the Laplace Transform of the source with Vm=1 is (Vm=1 for illustration simplicity):
Vs=s/(s^2+w^2)
so the current Is is just the source divided by the impedance:
Is=Vs/Z=(s/(s^2+w^2))/(R+s*L)
which simplified:
Is=s/((s^2+w^2)*(R+s*L))
so within a few seconds we've arrived at the basic solution.
And partial fraction expansion:
Is=(s*R+w^2*L)/((s^2+w^2)*(R^2+w^2*L^2))-(L*R)/((R+s*L)*(R^2+w^2*L^2))
and you can factor that any way you wish.
Partial fraction expansion isnt that hard once you do a few, but sometimes you'd have to have the numerical values of the components to proceed because the raw analytical form cant always be factored.
Ratchit offers some good advice here too...look for a table of Laplace Transforms and Operations as that will help sometimes...just dont depend on that to get all the answers all of the time. For example the sine and cos source transforms can be found in most LT tables, however you might not find those if they also contain a phase shift like cos(wt+ph).