# How to derive a differential equation of a first order circuit

1. Sep 14, 2012

### spoonjabba

Behaviour of any first-order circuit can be described by a first-order ordinary differential equation (often called the state equation) of the form :
dx/dt + αx = βy(t); x(0) = x0
where x is the state variable (usually the voltage across a capacitor or the current across an inductor), y(t) is a voltage or a current source, and the coefficients α, β are constants, depending on circuit elements. Assuming zero initial conditions in the circuit shown below:
1. Derive symbolic expressions for the coefficients α and β in the state equation.
2. If vs(t) = 35·u(t) V, calculate (numerically) the value of the inductor current x at t = 27 µs.

I understand how to derive the expression for a simple RC

dv/dt + v/RC = Vu(t)/dt which is the source

which is just kcl from the node above the capacitor

but for this circuit i have attached im a bit confuse as to where i should start? should i be zeroing sources? assuming the circuit is in steady state? mesh nodal thevenins nortorns ? any help will be appreciated.

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