Consider an electrical circuit comprising an inductor and a capacitor, with no external applied voltage, and no electrical resistance. The current, i, through the inductor is related to the charge, q, on the capacitor by the differential equation: where the inductance, L, and capacitance C, are positive constants.

L(di/dt)+q/c=0

(a)Given that the charge and the current are related through i=dq/dt derive a second order differential equation for q(t).

i know that the second order differential is

L(d2q/dt2)+q/c=0

please help me solve the problem

b)assuming that the initial charge q(0)=q0 and the initial current i(0)=i0, show that

q(t)=q0cos(wt)+(i0/w)sin(wt)

where w^2=1/LC

c) hence show that q(t) can be written as

q(t)=Acos(wt+fi)

where A and fi are to be determined

thanks for the help