A first-order differential equation $\frac{dy}{dx} = f(x, y)$ is classified as homogeneous if the function $f(x, y)$ can be expressed solely as a function of:
Options:
(A) $\frac{y}{x}$
(B) $x + y$
(C) $x \cdot y$
(D) $x - y$
Answer Key:
A
Step-by-Step Solution & Explanation
A function $f(x, y)$ is homogeneous of degree zero if $f(tx, ty) = t^0 f(x, y) = f(x, y)$. Setting $t = 1/x$, this becomes $f(1, y/x) = g(y/x)$. Hence, the right-hand side depends solely on the ratio $y/x$.