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: \[ f(x,y) = \varphi \left( \frac{y}{x} \right) \]

Answer 2

Step 1: Understand the Definition of a Homogeneous Function
A function \( f(x, y) \) is called homogeneous of degree \( n \) if it satisfies:
\[ f(tx, ty) = t^n f(x, y) \]
for all \( t \in \mathbb{R} \).

Step 2: What is a Homogeneous Differential Equation?
A first-order differential equation of the form:
\[ \frac{dy}{dx} = f(x, y) \]
is said to be homogeneous if the function \( f(x, y) \) is a homogeneous function of degree zero. This means:
\[ f(tx, ty) = f(x, y) \]
⇒ Degree \( n = 0 \)

Step 3: General Form of a Degree Zero Homogeneous Function
If \( f(x, y) \) is homogeneous of degree 0, then it can be expressed as a function of \( \frac{y}{x} \).
That is:
\[ f(x, y) = \varphi\left( \frac{y}{x} \right) \]

Final Answer:
The general form of \( f(x, y) \) for a homogeneous differential equation is:
\( \varphi\left( \frac{y}{x} \right) \)