Differential Equation of a LC Low Pass Filter
This circuit can be a low pass filter if the component values are selected properly. It has total 3 inductances+capacitors which cannot be reduced with series and parallel connections. This will finally need a 3rd order diff.equation. The equation will be linear.

If one wants to analyze this, he would write a group of 1st order linear differential equations. He presents the derivatives of capacitor voltages and the derivatives of the inductor currents with the input voltage, inductor currents and capacitor voltages. That's the state variable equation (group). It can be solved numerically in excel(=trace the changes of voltages and currents with small enough timesteps)

The state variable group can be converted to a nasty third order single variable diff equation. Believe, it will be complex if you want to have all component values as symbolic variables, not numerical.

General symbolic solution is so complex or useless or both that I wouldn't waste my day for it. A simple symbolic solution can be written with exponential matrix, but that's a gimmick, it has no numerical calculation value.

Laplace transforms can bring exact solutions with given boundary conditions with reasonable work.