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**Prove It** When you are solving this DE, I assume you are solving for $\displaystyle \displaystyle z$...

$\displaystyle \displaystyle \frac{dz}{dt} = \lambda z + g(t)$

$\displaystyle \displaystyle \frac{dz}{dt} - \lambda z = g(t)$

This is first order linear, so the Integrating Factor is $\displaystyle \displaystyle e^{\int{-\lambda \,dt}} = e^{-\lambda t}$.

Multiplying both sides by the Integrating Factor gives

$\displaystyle \displaystyle e^{-\lambda t}\,\frac{dz}{dt} - \lambda e^{-\lambda t}z = e^{-\lambda t}g(t)$

$\displaystyle \displaystyle \frac{d}{dt}(e^{-\lambda t}z) = e^{-\lambda t}g(t)$

$\displaystyle \displaystyle e^{-\lambda t}z = \int{e^{-\lambda t}g(t)\,dt}$

$\displaystyle \displaystyle z = e^{\lambda t}\int{e^{-\lambda t}g(t)\,dt}$.