Which of the following is a homogeneous differential equation? - Mathematics

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

MCQ

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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Homogeneous Differential Equations

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Q 17 Q 16 Q 1

Chapter 9: Differential Equations - Exercise 9.5 [Page 407]

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NCERT Mathematics Part 1 and 2 [English] Class 12

Chapter 9 Differential Equations
Exercise 9.5 | Q 17 | Page 407

2024-2025 (March) Official Board (with solutions)

Q 1. (ii) | 1 mark

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