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?
Answer:
v_b = ir + L(di/dt)
Step-by-step 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)