Which of the Following is a Homogeneous Differential Equation? - Mathematics

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

MCQ

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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Homogeneous Differential Equations

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Chapter 22: Differential Equations - MCQ [Page 143]

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RD Sharma Mathematics [English] Class 12

Chapter 22 Differential Equations
MCQ | Q 49 | Page 143

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