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Question
Which of the following is a homogeneous differential equation?
Options
(4x2 + 6y + 5) dy – (3y2 + 2x + 4) dx = 0
(xy) dx – (x3 + y3) dy = 0
(x3 + 2y2) dx + 2xy dy = 0
y2 dx + (x2 – xy – y2) dy = 0
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Solution
y2 dx + (x2 – xy – y2) dy = 0
Explanation:
Here y2 dx + (x2 – xy – y2) dy = 0
⇒ `dy/dx = - y^2/(x^2 - xy - y^2) = y^2/(y^2 + xy - x^2)`
Now, `f (x, y) = y^2/ (y^2 + xy - x^2)`
∴ `f lambda x, lambda y = (lambda^2 y^2)/(lambda^2 y^2 + (lambda x) (lambda y) - lambda^2x^2)`
`= lambda^0 (y^2/(y^2 + xy - x^2))`
`= lambda^0 f (x, y).`
∴ f (x, y) is homogeneous function of degree zero.
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