Question

Which of the following is a homogeneous differential equation? A. B. C. D.

Answer 1

Let, the function is F( x,y )and it will be homogeneous of degree n if F( λx,λy )= λ n F( x,y ).

The differential equation given in the option consider as follows.

y 2 dx+( x 2 −xy− y 2 )dy=0 dy dx = − y 2 ( x 2 −xy− y 2 ) dy dx = y 2 ( y 2 +xy− x 2 )

Let, F( x,y )= y 2 ( y 2 +xy− x 2 ) .

F( λx,λy )= ( λy ) 2 { ( λy ) 2 +( λx )( λy )− ( λx ) 2 } = λ 2 y 2 λ 2 ( y 2 +xy− x 2 ) = λ 0 F( λx,λy )

Hence, the differential equation is homogeneous and option (D) is correct.