Here is a problem sent by Neeraj
Let the current in the left inductor be and let the current in the right inductor be .
Recall that the voltage across an inductor is equal to its inductance multiplied by the time derivative of its current. The equations governing the circuit are:
The system matrix is:
The eigenvalues for this system are , and the associated eigenvectors are and . I used Wolfram alpha to find these.
The currents are therefore:
To solve for the constants, apply the initial conditions from time :
This results in .
The equations for the currents are then:
Interestingly, the inductors don't spend all of their energy as heat lost to the resistor. At , there is a circulating current of magnitude that only flows in a loop through the inductors.
The resistor current is:
From here, one can calculate the heat and charge as follows: