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Mathematics Question on Differential equations

Which of the following is a homogenous differential equation?

A

(4x+6y+5) dy−3(3y+2x+4) dx=0(4x+6y+5)\,dy-3(3y+2x+4)\,dx=0

B

(xy) dx−(x3+y3) dy=0(xy)\,dx-(x^3+y^3)\,dy=0

C

(x3+2y2) dx+2xy dy=0(x^3+2y^2)\,dx+2xy\,dy=0

D

y2 dx+(x2−xy−y2) dy=0y^2\,dx+(x^2-xy-y^2)\,dy=0

Answer

y2 dx+(x2−xy−y2) dy=0y^2\,dx+(x^2-xy-y^2)\,dy=0

Explanation

Solution

The correct answer is D:y2 dx+(x2−xy−y2) dy=0y^2\,dx+(x^2-xy-y^2)\,dy=0
Function F(x,y)F(x,y) is said to be the homogenous function of degree n,if
F(λx,λy)=λnF(X,Y)F(λx,λy)=λn F(X,Y)for any non-zero constant(λ)(λ).
Consider the equation given in alternative D:
y2dx+(x2−xy−y2)dy=0y^2dx+(x^2-xy-y^2)dy=0
⇒dydx=−y2x2−xy−y2=y2y2+xy−x2⇒\frac{dy}{dx}=\frac{-y^2}{x^2-xy-y^2}=\frac{y^2}{y^2+xy-x^2}
Let F(x,y)=y2y2+xy−x2.F(x,y)=\frac{y^2}{y^2+xy-x^2}.
⇒F(λx,λy)=(λy)2(λy)2+(λx)(λy)−(λx)2⇒F(λx,λy)=\frac{(λy)^2}{(λy)^2+(λx)(λy)-(λx)^2}
=λ2y2λ2(y2+xy−x2)=\frac{λ^2y^2}{λ^2(y^2+xy-x^2)}
=λ2y2λ2(y2+xy−x2)=\frac{λ^2y^2}{λ^2(y^2+xy-x^2)}
=λ°(y2y2+xy−x2)=λ°(\frac{y^2}{y^2+xy-x^2})
=λ°.F(x,y)=λ°.F(x,y)
Hence,the differential equation given in alternative D is a homogenous equation.