Ldi/dt + ir -Vcos wt=0 what technique do i use to solve?

Question

RL AC circuit

Answer 1

Use the integrating factor technique

First, get i' with no multiples:

i'+((-V*r*cos(w*t))/L)*i=0

The integrating factor is e^int(((-V*r*cos(w*t))/L)*dt)

which is e^(r*V*sin(w*t)/(L*w))

Multiply through by this, then recognize that the two left hand terms are a reverse product rule:

i'*[e^(r*V*sin(w*t)/(L*w))]+[e^(r*V*sin(w*t)/(L*w))*((-V*r*cos(w*t))/L)]*i=0

Compact the left two terms into a differential:

d/dt(e^(r*V*sin(w*t)/(L*w))*i)=0

Integrate both sides:

e^(r*V*sin(w*t)/(L*w))*i=C

Solve for i(t):

i(t)=C*e^(-r*V*sin(w*t)/(L*w))

Use your initial conditions on i and t to find the constant of integration. There you have it....

Answer 2

use the z transform