# Definition:Steady-State/First Order ODE

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## Definition

Consider the Decay Equation:

- $\dfrac {\d y} {\d x} = k \paren {y_a - y}$

where:

- $k \in \R: k > 0$
- $y = y_0$ at $x = 0$

which has the particular solution:

- $(1): \quad y = y_a + \paren {y_0 - y_a} e^{-k x}$

The term $y_a$ is known as the **steady-state** component of $(1)$.

### Electronics

The term is often seen in the context of the current in an $LR$ electric circuit $K$:

The electric current $I$ in $K$ is given by the equation:

- $I = \dfrac {E_0} R + \paren {I_0 - \dfrac {E_0} R} e^{-R t / L}$

The term $\dfrac {E_0} R$ is known as the **steady-state** part of $(1)$.