For this circuit model we need 3 basic component equations for the diode, capacitor, and inductor.
$$v_L=L\frac{di_L}{dt}$$
$$i_C=C\frac{dv_C}{dt}$$
$$i_D=I_o(e^{\frac{v_C}{kt/q}}-1)$$
With these equations, we can apply kirchoff’s Voltage law and rewrite the equations in state-space form.
$$\begin{bmatrix}
\frac{di_l}{dt} \\ \frac{dv_C}{dt}
\end{bmatrix}
=\begin{bmatrix}
\frac{v_C}{L}
\\ \frac{I_o(e^{\frac{v_C}{kt/q}}-1)-v_C/R-iL}C
\end{bmatrix}$$
[[State-Space model of a Power Converter]] – more complicated circuit
[[RLC Circuit Under Load]] – state-space example