Question

\( \frac{dy}{dx} = F(x, y) \) will be a homogeneous differential equation for which of the following functions ?
(i) \( F(x, y) = 3x + 2y \)
(ii) \( F(x, y) = \sin \frac{y}{x} + \log y - \log x \)
(iii) \( F(x, y) = e^{y/x} + 1 \)
(iv) \( F(x, y) = \sqrt{x^2 + y^2} - y \)

• A. (i) and (ii)
• B. (i), (ii) and (iii)
• C. (ii), (iii) and (iv)
• D. (ii) and (iii)

Hint:

A differential equation \( \frac{dy}{dx} = F(x,y) \) is homogeneous if every term in \( F(x,y) \) can be collectively expressed solely as a function of the ratio \( \frac{y}{x} \) or \( \frac{x}{y} \).

Answer 1

The Correct Option is C

Concept: A function \( F(x, y) \) is said to be homogeneous of degree \( n \) if substituting \( x \to \lambda x \) and \( y \to \lambda y \) results in \( F(\lambda x, \lambda y) = \lambda^n F(x, y) \). For a first-order differential equation \( \frac{dy}{dx} = F(x, y) \) to be classified as a homogeneous differential equation, the function \( F(x, y) \) must be a homogeneous function of degree 0, which means \( F(\lambda x, \lambda y) = F(x, y) \).

Step 1: Test function (i).
\( F(x, y) = 3x + 2y \) \[ F(\lambda x, \lambda y) = 3(\lambda x) + 2(\lambda y) = \lambda(3x + 2y) = \lambda^1 F(x, y) \] Here, the degree is 1, not 0. So, it does not form a homogeneous differential equation of the type \( \frac{dy}{dx} = F(x,y) \).

Step 2: Test function (ii).
\( F(x, y) = \sin \frac{y}{x} + \log y - \log x = \sin \frac{y}{x} + \log\left(\frac{y}{x}\right) \) \[ F(\lambda x, \lambda y) = \sin\left(\frac{\lambda y}{\lambda x}\right) + \log\left(\frac{\lambda y}{\lambda x}\right) = \sin\left(\frac{y}{x}\right) + \log\left(\frac{y}{x}\right) = \lambda^0 F(x, y) \] This is homogeneous of degree 0. Thus, (ii) is correct.

Step 3: Test function (iii).
\( F(x, y) = e^{y/x} + 1 \) \[ F(\lambda x, \lambda y) = e^{\frac{\lambda y}{\lambda x}} + 1 = e^{y/x} + 1 = \lambda^0 F(x, y) \] This is homogeneous of degree 0. Thus, (iii) is correct.

Step 4: Test function (iv).
\( F(x, y) = \sqrt{x^2 + y^2} - y \) \[ F(\lambda x, \lambda y) = \sqrt{(\lambda x)^2 + (\lambda y)^2} - \lambda y = \sqrt{\lambda^2(x^2 + y^2)} - \lambda y = \lambda \sqrt{x^2 + y^2} - \lambda y = \lambda^1 F(x, y) \] Here the function is homogeneous of degree 1. Let us check standard definitions: some conventions state that for \( \frac{dy}{dx} = F(x,y) \) to be homogeneous, \( F(x,y) \) can be written as a function of \( y/x \), meaning it must be degree 0. Let's re-verify option combinations. Since (ii) and (iii) are strictly degree 0, and option (D) is "(ii) and (iii)", it is the most standard correct choice under the strict degree-0 requirement for the derivative function expression.