Differential equations/Introduction to First Order Linear Differential Equations

A differential equation $y^{\prime }=f(t,y(t))$ is a first order differential equation. When placed in the form $y^{\prime }+p(t)y=g(t)$ 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:

• $y^{\prime }+3y=t$ where $p(t)=3$ and $g(t)=t$
• For $e^t{y}^{\prime }+3y=sin(t)$ we divide through by $e^t$ to place the equation in the proper form and $y^{\prime }+\frac{3}{e^t}y=\frac{sin(t)}{e^t}$ $p(t)=3e^{-t}$ and $g(t)=sin(t)e^{-t}$

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