Quiescent implies i₁, i₂ and their derivatives are zero for t = 0, ie

i₁(0) = i₂(0) = i₁'(0) = i₂'(0) = 0.

For loop 1:

For loop 2:

Substituting our result from (1) gives:

Taking Laplace transform:

Let

So

So

Taking Inverse Laplace:

So