Derive an expression for the total magnetic energy stored in two coils with inductances `L_(1)` and `L_(2)` and mutual inductance `M`, when the currents in the coils are `I_(1)` and `I_(2)`, respectively.

Explanation

To derive the expression for the total magnetic energy stored in two coils with inductances \( L_1 \) and \( L_2 \) and mutual inductance \( M \), when the currents in the coils are \( I_1 \) and \( I_2 \), respectively, we can follow these steps:

### Step 1: Define the Induced EMF in Each Coil
The induced electromotive force (EMF) in coil 1 due to its own current and the current in coil 2 can be expressed as:
\[ E_1 = -L_1 \frac{dI_1}{dt} - M \frac{dI_2}{dt} \]
Similarly, the induced EMF in coil 2 is given by:
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