Kapila is trying to find the general solution of the following differential equations.

(i) $x e^{\frac{x}{y}} dx - y e^{\frac{3x}{y}} dy = 0$
(ii) $(2x + 1) \frac{dy}{dx} = 3 - 2y$
(iii) $\frac{dy}{dx} = \sin x - \cos y$

Which of the above become variable separable by substituting $y = bx$, where $b$ is a variable?

only (i)

The correct answer is Option (1) → only (i) ##

Rewrite the differential equation $x e^{\frac{x}{y}} dx - y e^{\frac{3x}{y}} dy = 0$:

$\frac{x}{y} e^{\frac{x}{y}}dx-e^{\frac{3x}{y}}dy=0$

$\frac{dx}{dy} = \frac{e^{\frac{3x}{y}}}{\frac{x}{y} e^{\frac{x}{y}}}$

$\frac{dx}{dy} = \frac{e^{3}}{\frac{x}{y}}$

$\frac{dx}{dy} = \frac{1}{\frac{y}{x}e^3}$

$\frac{dx}{dy} = f\left(\frac{x}{y}\right)$

It is a homogeneous equation. This differential equation can be solved by using the method of variable separable by substituting $y = bx$.