What is the differential equation governing the growth of current in the circuit as a function of time after t

Complete question is;

What is the differential equation governing the growth of current in the circuit as a function of time after t = 0? express the right - hand side of the differential equation for di(t)/dt in terms of i(t), v_b, r, and l?

v_b = ir + L(di/dt)

Explanation:

I've attached the image of the circuit talked about in this question.

From the attached image, we can see that this circuit is an R-L circuit.

Since we want a differential equation for di(t)/dt that contains i(t), v_b, r, and L, we can start by finding v_b in R-L circuit which is;

v_b = v_r + v_i

where;

v_b is the voltage source.

v_r is the voltage across the resistance

v_i is the voltage across the inductance

Now, the voltage across the inductance could also be expressed as;

v_i = L(di/dt)

where;

L is the circuit inductance

Also, the voltage across the resistance could be as expressed as;

v_r = ir

Where;

r is the resistance

i is the current

Thus;

v_b = ir + L(di/dt)

if they have a finite resultant