Question

The differential equation \( \frac{dy}{dx} = F(x, y) \) will not be a homogeneous differential equation, if \( F(x, y) \) is:

• A. \( \cos x - \sin \left( \frac{y}{x} \right) \)
• B. \( \frac{y}{x} \)
• C. \( \frac{x^2 + y^2}{xy} \)
• D. \( \cos^2 \left( \frac{x}{y} \right) \)

Hint:

A function \( F(x, y) \) is homogeneous if it can be expressed entirely in terms of \( \frac{y}{x} \) or \( \frac{x}{y} \). Check each term carefully to determine homogeneity.

Answer 1

The Correct Option is A

Step 1: Definition of a homogeneous differential equation.
A differential equation \( \frac{dy}{dx} = F(x, y) \) is homogeneous if \( F(x, y) \) can be written as a function of \( \frac{y}{x} \) (or equivalently \( \frac{x}{y} \)). Step 2: Analyze each option.
We check whether \( F(x, y) \) in each option can be expressed in terms of \( \frac{y}{x} \) or \( \frac{x}{y} \). - (A) \( F(x, y) = \cos x - \sin \left( \frac{y}{x} \right) \): The term \( \cos x \) depends only on \( x \), and it cannot be expressed as a function of \( \frac{y}{x} \). Hence, \( F(x, y) \) is not homogeneous. - (B) \( F(x, y) = \frac{y}{x} \): Clearly, \( \frac{y}{x} \) is already in the required form, so it is homogeneous. - (C) \( F(x, y) = \frac{x^2 + y^2}{xy} \): Simplify: \[ F(x, y) = \frac{x^2}{xy} + \frac{y^2}{xy} = \frac{x}{y} + \frac{y}{x}. \] Both terms \( \frac{x}{y} \) and \( \frac{y}{x} \) are functions of \( \frac{y}{x} \), so \( F(x, y) \) is homogeneous. - (D) \( F(x, y) = \cos^2 \left( \frac{x}{y} \right) \): The term \( \cos^2 \left( \frac{x}{y} \right) \) depends solely on \( \frac{x}{y} \), so \( F(x, y) \) is homogeneous. Step 3: Conclusion.
The only function that is not homogeneous is: \[ \boxed{\cos x - \sin \left( \frac{y}{x} \right)}. \]