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}, we shall use method of separating the variables.

Given differential equation of the form

\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)

By separating the variable technique we must keep the terms dy and dx in the numerators with their respective functions.
Now integration 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 logarithm from both sides of the above equation, we get

y =  xc

Which is the required solution of the given differential equation.