Question

What is a homogeneous differential equation?

Answer 1

Homogeneous differential equation.

• The homogeneous differential equation is a differential equation, in which the functions $f\left(x,y\right)$ and $g\left(x,y\right)$ are of the same degree.
• For example, differential equation $\frac{dy}{dx}=f(x,y)$ is a homogeneous differential equation, if the function $f(x,y)$ is a homogeneous function.
• $f\left(x,y\right)$ and $g\left(x,y\right)$ are the homogeneous functions of the same degree.
• The homogeneous function of degree $n$ is of the form: $f(x,y)=x^n\times g\left(\frac{y}{x}\right)\mathrm{or}f(x,y)=y^n\times h\left(\frac{y}{x}\right)$.

Thus, the homogeneous differential equation is a differential equation, in which the functions $f\left(x,y\right)$ and $g\left(x,y\right)$ are of the same degree.