For the homogeneous differential equation $\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}$, applying the substitution $y = vx$ converts it into the separable differential equation:
Options:
(A) $\frac{2v}{1 - v^2} dv = \frac{dx}{x}$
(B) $\frac{2v}{1 + v^2} dv = \frac{dx}{x}$
(C) $\frac{1}{1 - v^2} dv = \frac{dx}{x}$
(D) $\frac{v}{1 - v} dv = \frac{dx}{x}$
Answer Key:
A
Step-by-Step Solution & Explanation
Let $y = vx$. Then $\frac{dy}{dx} = v + x \frac{dv}{dx}$.
Substitute into the differential equation:
$v + x \frac{dv}{dx} = \frac{x^2 + (vx)^2}{2x(vx)} = \frac{1 + v^2}{2v}$
$x \frac{dv}{dx} = \frac{1 + v^2}{2v} - v = \frac{1 + v^2 - 2v^2}{2v} = \frac{1 - v^2}{2v}$
Separating variables:
$\frac{2v}{1 - v^2} dv = \frac{dx}{x}$
Thus, option A is correct.