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The above equations on application of Laplace Transform become
\begin{align*}
sL_1\hat{i}_{L1}(s) &=
\!\begin{aligned}[t]
&D\cdot \hat{v}_{C1}(s) + D\cdot \hat{v}_{C2}(s) - D' \cdot \hat{v}_g(s) \\
&\qquad + (C_{C1} + V_{C2} - V_g + V_D)\cdot \hat{d}(s)
\end{aligned}
\\[1\jot]
sL_2\hat{i}_{L2}(s) &=
\!\begin{aligned}[t]
&{-}D' \cdot \hat{v}_{C1}(s) + D\cdot \hat{v}+{C2}(s) + D' \cdot \hat{v}_g(s) \\
&\qquad + (C_{C1} + V_{C2} - V_g + V_D)\cdot \hat{d}(s)