Show that dydxx2xyy2x2 is homogeneousSolutionx2 dy x2xyy2 d

Show that dy/dx=(x^2+xy+y^2)/x^2 is homogeneous.

Solution

x^2 dy - (x^2+xy+y^2) dx = 0

x^2 and x^2+xy+y^2 are homogeneous functions of the same degree 2.

Therefore, dy/dx=(x^2+xy+y^2)/x^2 is homogeneous.

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Definition of homoegeneous diffeential equations:

A homogeneous differential equation is an equation of the form M(x, y)dx + N(x, y)dy = 0

where M and N are homogeneous functions of hte same degree.