The current I(t) in the electric circuit by the differential...

The current I(t) in the electric circuit by the differential equation L dI/dt + RI = E(t) Find an expression for the current in a circuit where the resistance is 12 ?, the inductance is 4 H, a battery gives a constant voltage of 60 V, and the switch is turned on when t = 0. What is the limiting value of the current?

Transcript

00:01 Hi, here for the given question the equation which we are given is l di upon dt plus ri is equal to e of t.

00:11 So, here in our case we know that it represents the current and here r is the resistance and l is the inductance.

00:19 We are given that r is equal to 12 ohm, l is equal to 4h and the voltage e of t is equal to 60 volt and t is equal to 0.

00:30 So, substituting the value over here we have 4 di upon dt plus 12i is equal to 60.

00:40 Simplifying this further di upon dt plus 3i is equal to 15.

00:45 So, here let this be our equation 1.

00:47 This is a linear equation.

00:49 So, here we can find the integrating factor if is equal to integration of e to the power integration of 3dt.

00:55 So, this is equal to e to the power 3t.

00:58 So, the solution can be written as here we have value as e3t into di upon dt plus 3 times e to the power 3t i is equal to 15 into e to the power 3t.

01:12 So, here we have d by dt of e to the power 3t i and here this is equal to 15 times e to the power 3t.

01:22 So, here we will integrate on both the sides with respect to dt.

01:27 So, here in our case now on simplifying this we have e to the power 3t i equals to 15 times e to the power 3t upon 3 plus c where c is an arbitrary constant.

01:39 So, here this can be again written as e to the power 3t i is equal to 5 times e to the power 3t plus c.

01:46 So, i can be written as 5 plus c times e to the power minus 3t...

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