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I have seen in multiple books the following fact:

Suppose we have an n-th order \$RLC\$ LTI circuit with and independent volatage source \$x(t)\$. Then a voltage \$y(t)\$ of some node in the circuit can be represented by the following differential equation:

$$\frac{d^n y}{d^n t}+a_{n-1}\frac{d^{n - 1} y}{d^{n - 1} t} + ... + a_{1}\frac{d y}{dt} + a_0 y= b_m\frac{d^m x}{d^m t}+b_{m-1}\frac{d^{m - 1} x}{d^{m - 1} t} + ... + b_{1}\frac{d x}{dt} + b_0 x$$

Nodal analysis gives a system of second order (due to \$RLC\$ relations) equations, but I can't figure out how to in general isolate a single voltage node \$y\$ variable from it. It is easier to solve this general system by transforming it to a system of first order equations, but I want to do this by isolating a single variable into an n-th order ODE.

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  • \$\begingroup\$ "Easy way": Write the node equations in the Fourier domain in matrix form. Solve the matrix equation (by Gauss-Jordan elimination or whatever method you like). Convert the Fourier domain solution back into a differential equation. \$\endgroup\$
    – The Photon
    Commented Jan 7 at 17:16

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