Which of the following is a homogeneous diffe | vedclass

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(D) differential equation $\frac{dy}{dx} = f(x, y)$ is homogeneous if $f(x, y)$ is a homogeneous function of degree $0$,meaning $f(\lambda x, \lambda

Vedclass · Accepted Answer

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

Vedclass · Answer

(D) differential equation $\frac{dy}{dx} = f(x, y)$ is homogeneous if $f(x, y)$ is a homogeneous function of degree $0$,meaning $f(\lambda x, \lambda y) = \lambda^0 f(x, y) = f(x, y)$.Let us check option $D$: $y^2 dx + (x^2 - xy - y^2)dy = 0$.This can be written as $\frac{dx}{dy} = -\frac{x^2 - xy - y^2}{y^2} = \frac{-x^2 + xy + y^2}{y^2}$.Let $F(x, y) = \frac{-x^2 + xy + y^2}{y^2}$.Substituting $x = \lambda x$ and $y = \lambda y$:$F(\lambda x, \lambda y) = \frac{-(\lambda x)^2 + (\lambda x)(\lambda y) + (\lambda y)^2}{(\lambda y)^2} = \frac{\lambda^2(-x^2 + xy + y^2)}{\lambda^2 y^2} = \frac{-x^2 + xy + y^2}{y^2} = F(x, y)$.Since $F(\lambda x, \lambda y) = \lambda^0 F(x, y)$,the function is homogeneous of degree $0$,and thus the differential equation is homogeneous.

Which of the following is a homogeneous differential equation?

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