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.
Kirk R Wayland
September 14 @ 8:20 am
dy/dx = x + y, y(0) = 1. My answer is y = x + 1 + Ce^x, correct?
A. Wolf
May 9 @ 9:36 pm
The equation y = xc tells you nothing because c can take any value. It is literally as useful as saying 0 = 0.
It doesn’t make any sense to manipulate dy/dx = y/x in the first place if you understand what that equation actually means, and this tutorial does not explain that to the reader. This is not a helpful tutorial.