# Solve Differential Equation dy/dx=y/x

In this tutorial we shall evaluate the simple differential equation of the form $\frac{{dy}}{{dx}} = \frac{y}{x}$, and we shall use the method of separating the variables.

The differential equation of the form is given as

$\frac{{dy}}{{dx}} = \frac{y}{x}$

Separating the variables, the given differential equation can be written as

$\frac{1}{y}dy = \frac{1}{x}dx\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right)$

With the separating the variable technique we must keep the terms $dy$ and $dx$ in the numerators with their respective functions.

Now integrating both sides of the equation (i), we have
$\int {\frac{1}{y}dy = \int {\frac{1}{x}dx} }$

Using the formula of integration $\int {\frac{1}{x}dx = \ln x + c}$, we get
$\begin{gathered} \ln y = \ln x + \ln c \\ \Rightarrow \ln y = \ln xc \\ \end{gathered}$

Cancelling the logarithm from both sides of the above equation, we get
$y = xc$

This is the required solution of the given differential equation.