# Differential equations/Introduction to First Order Linear Differential Equations

A differential equation ${\displaystyle y'=f(t,y(t))}$ is a first order differential equation. When placed in the form ${\displaystyle y'+p(t)y=g(t)}$ where ${\displaystyle p(t)}$ and ${\displaystyle g(t)}$ are functions defined on an interval ${\displaystyle a is called a first order linear differential equation.

For example:

• ${\displaystyle y'+3y=t}$ where ${\displaystyle p(t)=3}$ and ${\displaystyle g(t)=t}$
• For ${\displaystyle e^{t}y'+3y=sin(t)}$ we divide through by ${\displaystyle e^{t}}$ to place the equation in the proper form and ${\displaystyle y'+{\frac {3}{e^{t}}}y={\frac {sin(t)}{e^{t}}}}$ ${\displaystyle p(t)=3e^{-t}}$ and ${\displaystyle g(t)=sin(t)e^{-t}}$

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