solve transient circuit with diode

Hello to everyone!

I'm dealing with the mathematical model and its solution for a simple circuit in presence of a non linear component like the diode in Matlab (I'm able to solve the problem via simulink).

Usually, when I would like to study the transient of a circuit, I use ode45 and the state-space model.

Now, I'm trying to build an equation set able to describe the time evolution of the circuit in the picture with a diode and a sinusoidal voltage source.

In order to treat the diode, I'm using the simplified Shockley equation i.e. (where and are constants). This implies to deal with the exponential function or a logarithmic function in case I would prefer to describe the diode by means of the voltage .

Unfortunately, I'm not able to go ahead from this point as I cannot write a set of ODEs and I never dealt with non linear components in this context.

Could somebody give me some hint?

Thank you!