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?
Answer & explanation
Correct answer: option 1
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 $y = bx$.