Question

If \( \frac{dy}{dx} = f(x,y) \) is a homogeneous differential equation, then the general form of \( f(x,y) \) is:

• A. \( x^n \varphi \left( \frac{y}{x} \right), \ n \neq 1 \)
• B. \( y^n \varphi \left( \frac{x}{y} \right), \ n \neq 1 \)
• C. \( \varphi \left( \frac{y}{x} \right) \)
• D. \( K \varphi \left( f(x,y) \right), \ n \neq 1 \)

Hint:

For a homogeneous differential equation, always express the function \( f(x,y) \) in terms of \( \frac{y}{x} \), which simplifies the process of solving the equation.

Answer 1

The Correct Option is C

A homogeneous differential equation of the form \( \frac{dy}{dx} = f(x,y) \) is said to be homogeneous if the function \( f(x, y) \) can be written as a function of \( \frac{y}{x} \). This means that \( f(x, y) \) depends on the ratio of \( y \) and \( x \), and hence can be expressed in the form \( \varphi \left( \frac{y}{x} \right) \), where \( \varphi \) is some function. Thus, the general form of \( f(x, y) \) for a homogeneous differential equation is: \[ f(x,y) = \varphi \left( \frac{y}{x} \right) \]