Differential equations/Introduction to First Order Linear Differential Equations

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

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

Method