circuit equations derivation problem
Why am I getting the sign of that IL term wrong?


The voltage across an inductor is related to the current through the inductor as follows:

\$v_L = L \dfrac{di_L}{dt} \$

where the reference direction for the current is into the positive labelled terminal.



Since:

\$v_L = V_2 \$

it immediately follows that

\$\dfrac{di_L}{dt} = \dfrac{V_2}{L} \$