Question

Which differential equation is called a homogeneous differential equation?

Options

• \[\frac{dy}{dx}=\frac{f_{1}(x,y)}{f_{2}(x,y)},\] where \[f_{1}(x,y)\] and \[f_{2}(x,y)\] are homogeneous functions of \[x\] and \[y\] of the same degree.

• \[\frac{dy}{dx}=\frac{f_{1}(x)}{f_{2}(y)},\] where \[f_{1}(x)\] and \[f_{2}(y)\] are non-zero constants.

• \[\frac{dx}{dy}=\frac{f_{1}(x,y)}{f_{2}(x,y)},\] where \[f_{1}(x,y)\] is constant.

• \[\frac{dy}{dx}=f_{1}(x,y)+f_{2}(x,y),\] where \[f_{1}(x,y)\] and \[f_{2}(x,y)\] have different degrees.

Answer 1

A homogeneous differential equation has the form \[\frac{dy}{dx}=\frac{f_{1}(x,y)}{f_{2}(x,y)}\]. Both \[f_{1}(x,y)\] and \[f_{2}(x,y)\] must be homogeneous functions of \[x\] and \[y\] of the same degree.