Solutions of Differential Equations

The value of the dependent variable which along with its derivatives is involved in the differential equation and satisfies the differential equation is called the solution of that equation.

For example, consider the first order differential equation:
$\frac{{dy}}{{dx}} – y = 0\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right)$

Let $y = c{e^x}$, then $\frac{{dy}}{{dx}} = c{e^x}$. Putting these values of $y$ and $\frac{{dy}}{{dx}}$ in equation (i), we have
$\begin{gathered} c{e^x} – c{e^x} = 0 \\ \Rightarrow 0 = 0 \\ \end{gathered}$

This shows that $y = c{e^x}$ satisfies equation (i), so $y = c{e^x}$ is the solution of equation (i). Since $c$ is an arbitrary constant, this solution is called the general solution of the differential equation. In fact, the general solution of a second order differential equation has two arbitrary constants.

Among the simpler differential equations of the first order are those in which the two variables involved in the differential equations are separated. The solution of such a differential equation can be obtained by simple integration.