 # What Are First Order Differential Equations?

A first order differential equation is an equation where you have both a function $y$ and its derivative ${y}^{\prime }$. It is called “first order”, because the highest order derivative you can have is the first derivative.

## Verifying If a Given Function Is a Solution

You can verify if a given function is a solution to the differential equation. You insert the function and its derivative into the equation and see if you get the same expression on both sides of the equation. This involves finding the first derivative of the function, so that you can insert it into the differential equation.

Example 1

Check if ${y}_{0}=x\cdot {e}^{x}$ is a solution to the differential equation ${y}^{\prime }-y={e}^{x}$

Find the derivative of ${y}_{0}$:

 ${y}_{0}^{\prime }={\left(x\cdot {e}^{x}\right)}^{\prime }={e}^{x}+x\cdot {e}^{x}$

Then insert the expressions for $y$ and ${y}^{\prime }$ into the differential equation:

$\begin{array}{llll}\hfill \text{L.S.}& ={y}^{\prime }-y\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & ={e}^{x}+x\cdot {e}^{x}-x\cdot {e}^{x}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & ={e}^{x}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\text{R.S.}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{lll}\hfill \text{L.S.}={y}^{\prime }-y={e}^{x}+x\cdot {e}^{x}-x\cdot {e}^{x}={e}^{x}=\text{R.S.}& \phantom{\rule{2em}{0ex}}& \hfill \end{array}$

The left-hand side and the right-hand side are the same, so you know that ${y}_{0}=x\cdot {e}^{x}$ is a solution!