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Which of the following is a homogeneous differential equation?

Question:

Which of the following is a homogeneous differential equation?

A. $(4 x+6 y+5) d y-(3 y+2 x+4) d x=0$

B. $(x y) d x-\left(x^{3}+y^{3}\right) d y=0$

C. $\left(x^{3}+2 y^{2}\right) d x+2 x y d y=0$

D. $y^{2} d x+\left(x^{2}-x y-y^{2}\right) d y=0$

Solution:

Function $\mathrm{F}(x, y)$ is said to be the homogenous function of degree $n$, if $\mathrm{F}(\lambda x, \lambda y)=\lambda^{n} \mathrm{~F}(x, y)$ for any non-zero constant $(\lambda)$.

Consider the equation given in alternativeD:

$y^{2} d x+\left(x^{2}-x y-y^{2}\right) d y=0$

$\Rightarrow \frac{d y}{d x}=\frac{-y^{2}}{x^{2}-x y-y^{2}}=\frac{y^{2}}{y^{2}+x y-x^{2}}$

Let $F(x, y)=\frac{y^{2}}{y^{2}+x y-x^{2}}$.

$\Rightarrow F(\lambda x, \lambda y)=\frac{(\lambda y)^{2}}{(\lambda y)^{2}+(\lambda x)(\lambda y)-(\lambda x)^{2}}$

$=\frac{\lambda^{2} y^{2}}{\lambda^{2}\left(y^{2}+x y-x^{2}\right)}$

$=\lambda^{0}\left(\frac{y^{2}}{y^{2}+x y-x^{2}}\right)$

$=\lambda^{0} \cdot F(x, y)$

Hence, the differential equation given in alternative D is a homogenous equation.

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