urgent please help

Question

Suppose that in the simple RL circuit, the resistance is 20W and the inductance is 1 H. If a generator supplies a voltage of 𝐸(𝑡) = 60 sin 40𝑡 volts, and i(0) = 1A. Find i(t).

Answer 1

L(di/dt) + Ri = E(t)

L = 1, R = 20, and E(t) = 60sin(40t)

So, di/dt + 20i = 60sin(40t)

This is a first order linear differential equation. Multiply both sides by the integrating factor e^∫20dt = e^20t to get

e^20t(di/dt) + 20e^20ti = 60e^20tsin(40t)

The left side is the derivative of e^20ti.

Integrate both sides to get e^20ti = 60∫e^20tsin(40t)dt

Evaluate the integral on the right side using integration by parts twice to get:

e^20ti = -(6/5)e^20tcos(40t) + (3/5)e^20tsin(40t) + C

So, i(t) = -(6/5)cos(40t) + (3/5)sin(40t) + Ce^-20t

Since i(0) = 1, we have C = 11/5

So, i(t) = (11/5)e^-20t - (6/5)cos(40t) + (3/5)sin(40t)