A function \(f(x,y)\) is said to be homogenous of degree \(n\) if degree of each term of the function \(f(x,y)\) is  \(n\). For example \(f(x,y) = {x^3} + {x^2}y + {y^3}\)is a homogeneous function of degree 3. Alternately a function \(f(x,y)\) of degree \(n\)can be expressed as \(f(\lambda x,\,\lambda y) = {\lambda ^n}f(x,\,y)\). 

In homogeneous differential equation, \(\cfrac{{dy}}{{dx}} = \cfrac{{f(x,y)}}{{g(x,y)}}\), where \(f(x,y)\) and \(g(x,y)\) are homogeneous functions of same degree.

To solve such types of equations we put \(y = vx\)

\[ \Rightarrow \frac{{dy}}{{dx}} = v + x\frac{{dv}}{{dx}}\] The given equation then reduces to \[v + x\frac{{dv}}{{dx}} = F(v)\] \[ \Rightarrow x\frac{{dv}}{{dx}} = F(v) - v\]

Hence, the variables are separated in terms of \(v\) and \(x\).  

Example: Solve the differential equation \(\cfrac{{dy}}{{dx}} = \cfrac{{{x^2} + xy}}{{{x^2} - xy}}\)

Solution: This is a homogeneous equation, so put \(y = vx\)

\[ \Rightarrow \frac{{dy}}{{dx}} = v + x\frac{{dv}}{{dx}}\]

\[ \Rightarrow v + x\frac{{dv}}{{dx}} = \frac{{1 + v}}{{1 - v}}\]

\[ \Rightarrow \frac{{1 - v}}{{1 + {v^2}}}dv = \frac{1}{x}dx\]

Integrating, we have,

\[\int {\left( {\frac{1}{{1 + {v^2}}}} \right)dv}  - \frac{1}{2}\int {\left( {\frac{{ - 2v}}{{1 + {v^2}}}} \right)dv = \int {\frac{1}{x}dx} } \]

\[ \Rightarrow {\tan ^{ - 1}}v - \frac{1}{2}\log (1 + {v^2}) = \log x + c\]

\[ \Rightarrow {\tan ^ - }\left( {\frac{y}{x}} \right) - \frac{1}{2}\log \left( {1 + \frac{{{y^2}}}{{{x^2}}}} \right) = \log x + c\]