Question

Which of the following is a homogeneous differential equation?

• A. $(4x+6y+5)dy+(3y+2x+4)dx=0$
• B. $\left(xy\right)dx+(x^3+y^3)dy=0$
• C. $(x^3+2y^2)dx+2xydy=0$
• D. $y^{2}dx+(x^2+xy+y^2)dy=0$

Answer 1

The correct option is D $y^{2}dx+(x^2+xy+y^2)dy=0$

A) $(4x+6y+5)dy+(3y+2x+4)dx=0$

$M\left(x\right)=4x+6y+5$

$N\left(x\right)=3y+2x+4$

If we put $x=2x,y=2y$

$M(2x,2y)=z(4x+6y)+5$

$N(2x,2y)=z(3y+2x)+4$

$M(2x,2y)\ne z^{n}M(x,y)$

$N(2x,2y)\ne z^{n}N(x,y)$

So, it is not a homogeneous.

B) $\left(xy\right)dx+(x^2+y^3)dy=0$

$M(x,y)=x^3+y^3$

$N(x,y)=xy$

$M(2x,2y)=z^2(x^3+y^3)=z^3\left(M\right(x,y)$

$N(2x,2y)=z^2\left(xy\right)=z^{2}N(x,y)$

but degree is not same

So, it is not homogeneous equation.

C) $(x^3+2y^2)dx+2xydy=0$

$M(x,y)=2xy$

$M(2x,2y)=2z^{2}xy=z^{2}M(x,y)$

$N(x,y)=x^3+2y^2$

$N\left(\right(2x,2y)=z^3{x}^3+2z^2{y}^2\ne z^3,z^{n}N(x,y)$

So, it is not a homogenous.

D) $y^{2}dx+(x^2+xy+y^2)dy=0$

$m(x,y)=X^2+xy+y^2$

$n(x,y)=Y^2$

$m(2x,2y)=z^2(x^2+xy+y62)=z^{2}m(x,y)$

$N(2x+2y)=z^2{y}^2=z^{2}N(x,y)$

So, it is a homogeneous equation.