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.