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Question
Which of the following is a homogeneous differential equation?
Options
`(4x + 6y + 5) dy - (3y + 2x + 4) dx` = 0
`xy dx - (x^3 + y^3) dy` = 0
`(x^3 + 2y^2) dx + 2xy dy` = 0
`y^2 dx + (x^2 - xy - y^2) dy` = 0
Solution
`y^2 dx + (x^2 - xy - y^2) dy` = 0
Explanation:
Consider the differential' equation,
`y^2dx + (x^2 - xy - y^2) dy` = 0
∴ `(dx)/(dy) = (-y^2)/(x^2 - xy - y^2) = (y^2)/(x^2 + xy - 9x^2) = f(x, y)`
`f(x, y) = y^2/(x^2 + xy - x^2)`
Replacing `x` by `lambdax` and `y` by `lambday`
`f(lambdax, lambday) = (lambday)^2/((lambdax)^2 + (lambdax)(lambday) - (lambdax)^2`
= `lambda^circ (lambda^2y^2)/(lambda^2x^2 + lambda^2xy - lambda^2x^2)`
= `(lambda^2y^2)/(lambda^2(y^2 + xy - x^2))`
= `lambda0 (y^2/(y^2 + xy - x^2))`
= `lambda^circ f(x, y)`
∴ `f(x, y)` is the homogeneous function of degree zero.
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