Which of the following is a homogeneous differential equation?

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Solution

Q.
Which of the following is a homogeneous differential equation

1 $(4 x + 6 y + 5) d y - (3 y + 2 x + 4) d x = 0$

2 $x y d x - (x^{3} + y^{3}) d x = 0$

3 $(x^{3} + 2 y^{2}) d x + 2 x y d y = 0$

4 $y^{2} d x + (x^{2} - x y - y^{2}) d y = 0$

Ans.
(4)

Ans. $y^{2} d x + (x^{2} - x y - y^{2}) d y = 0$

Explanation:

By equation, $y^{2} d x + (x^{2} - x y - y^{2}) d y = 0$

$∴$ $\frac{d y}{d x} = \frac{- y^{2}}{x^{2} - x y - y^{2}}$

Power of numerator and denominator are of same order. Hence, it is a homogeneous differential equation.

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