Ex3. Find the charge on the capacitor and the current in the given LRC series circuit. Find the maximum charge on the capacitor.

Note h=henrys, =ohms, f=farads, C=coulombs, A=amperes, V=volts.

Solution:

The equation is . The characteristic equation of the homogeneous DE has –50 as a double root. The general solution of the homogeneous equation is

Using undetermined coefficients, we could assume a particular solution to be of constant form A, and A would have first and second derivative both zero, hence A=30/2500 or A=.012

The general solution to the non-homogeneous equation is

The initial conditions q(0)=0 and q’(0)=2 produce the values for C₁ and C₂ but first we need q’(t):

Substituting our IC gives the system of equations

from which we obtain C₁=-.012 and C₂=1.4.

The solution to the IVP is:

The maximum charge on the capacitor will occur when i=q’=0, and solving i=0 we get t=1/35. The maximum charge on the capacitor occurs when t=1/35, so we find the charge by finding q(1/35).