Show that the system of differential equations for the charg

Show that the system of differential equations for the charge on the capacitor q(t) and the current i3 (t) in the electrical network shown below is R1 dq/dt+1/Cq+R1 i3 = E(t) Ldi/dt + R2i3 -1/Cq=0 Use the Laplace method to find the charge on the capacitor when L=1 h, R1=3Ohm, R2=1 Ohm, C=1 E(t) = , i=(0)=0,and q(0)=0.

Solution

(a)

Let the E containing loop to be 1 and L contaiting loop to be 2.

In loop 1 apply the Kirchoff law :-

E -i₁R₁ -(q/c) =0

i_{1 =} i₂ + i₃

i₂ = dq/dt

Manage all the result to get the equation.

R₁dq/dt +q/c + R₁ i₁ =E(t)

(b)

In loop 2 apply the Kirchoff law :-

- Ldq/dt - R₂ i₃+q/c +  =0