Solve the differential equation of L-C circuit and obtain the expression of current.

Explanation

The differential equation of L-C circuit is,
`(d^(2)q)/( dt^(2))+(1)/(LC) q = 0`
The solution of the equation is,
`q= q_(m) cos ( omega_(0) t+ phi)` …(1)
where `q_(m)` is the maximum value of q and `phi` is phase constant.
At t= 0, `q= q_(m)`
`:. q_(m) = q_(m) cos phi`
`:. 1 = cos phi`
`:. phi = 0^(@)`
`:.` From equation (1),
`q = q_(m) cos ( omega_(0) t)`
Differentiating w.r.t. time t,
`(dq)/(dt) = q_(m) omega_(0)( - sin omega_(0)t)`
but `(dq)/(dt) = -1`
`:. -I = - q_(m) omega_(0) sin omega_(0) t `
`:. I = I_(m) sin omega_(0) t `
where,` q_(m) omega _(0) =I_(m) `
`:. I = I_(m) sin omega_(0) t`