Question

Which of the following is a homogenous differential equation?

• A. \((4x+6y+5)\,dy-3(3y+2x+4)\,dx=0\)
• B. \((xy)\,dx-(x^3+y^3)\,dy=0\)
• C. \((x^3+2y^2)\,dx+2xy\,dy=0\)
• D. \(y^2\,dx+(x^2-xy-y^2)\,dy=0\)

Answer 1

The Correct Option is D

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