# Math Help - Help With differntial equation

1. ## Help With differntial equation

i am newbie to this topic at the moment

thanks

2. Hello, master_2m8!

$\frac{dy}{dt} \:=\:\frac{2}{100}y - 50$

We have: . $\frac{dy}{dt} - \frac{1}{50}y \:=\:-50$

Integrating factor: . $I \:=\:e^{\int\text{-}\frac{1}{50}\,dt} \:=\:e^{\text{-}\frac{1}{50}t}$

Multiply by $I\!:\;\;e^{\text{-}\frac{1}{50}t}\frac{dy}{dt} -\frac{1}{50}e^{\text{-}\frac{1}{50}t}y \;=\;-50\,e^{\text{-}\frac{1}{50}t}$

We have: . $\frac{d}{dt}\left(e^{\text{-}\frac{1}{50}t}y\right) \:=\: -50\,e^{\text{-}\frac{1}{50}t}$

Integrate: . $e^{\text{-}\frac{1}{50}t}y \;=\;-50\!\int e^{\text{-}\frac{1}{50}t}dt$

. . . . . . . . $e^{\text{-}\frac{1}{50}t}y \;=\;2500\,e^{\text{-}\frac{1}{50}t} + C$

Multiply by $e^{\frac{1}{50}t}\!:\quad\boxed{y \;=\;2500 + C\,e^{\frac{1}{50}t}}$