Question

What is the meaning of a homogeneous equation?

Answer 1

Homogeneous Equation:

A differential equation of the form $\frac{dy}{dx}=f\left(x,y\right)$ is said to be homogeneous if $f\left(x,y\right)$ is a homogeneous function of degree $0$.

Whereas the function $f\left(x,y\right)$ is to be homogeneous function of degree $n$ if for any non-zero constant $\lambda$, $f\left(\lambda x,\lambda y\right)={\lambda }^{n}f\left(x,y\right)$

For example: $\frac{dy}{dx}=\frac{x^2-4y^2}{3xy-5x^2}$is a homogeneous differential equation.

We have $f\left(x,y\right)=\frac{x^2-4y^2}{3xy-5x^2}$

For any non-zero constant $\lambda$, we have

$\begin{array}{rcl}f\left(\lambda x,\lambda y\right)& =& \frac{{\left(\lambda x\right)}^2-4{\left(\lambda y\right)}^2}{3\left(\lambda x\right)\left(\lambda y\right)-5{\left(\lambda x\right)}^2}\\ & \Rightarrow & f\left(\lambda x,\lambda y\right)=\frac{{\lambda }^2\left(x^2-4y^2\right)}{{\lambda }^2\left(3xy-5x^2\right)}\\ & \Rightarrow & f\left(\lambda x,\lambda y\right)={\lambda }^0\left(\frac{x^2-4y^2}{3xy-5x^2}\right)\\ & \Rightarrow & f\left(\lambda x,\lambda y\right)={\lambda }^{0}f\left(x,y\right)\\ & & \end{array}$

So, $f\left(x,y\right)$ is a homogeneous function of degree $0$.

Thus, $\frac{dy}{dx}=\frac{x^2-4y^2}{3xy-5x^2}$is a homogeneous differential equation.