Problem 04 | Separation of Variables

$\dfrac{2y \, dx}{xy} = \dfrac{3x \, dy}{xy}$

$\dfrac{2 \, dx}{x} = \dfrac{3 \, dy}{y}$

$\displaystyle 2 \int \dfrac{dx}{x} = 3 \int \dfrac{dy}{y}$

$2 \ln x = 3 \ln y + \ln c$

$2 \ln x - 3 \ln y = \ln c$

$\ln x^2 - \ln y^3 = \ln c$

$\ln \dfrac{x^2}{y^3} = \ln c$
 

Therefore,
$\dfrac{x^2}{y^3} = c$
 

when x = 2, y = 1
$\dfrac{2^2}{1^3} = c$

$c = 4$
 

Thus,
$\dfrac{x^2}{y^3} = 4$

$y^3 = \dfrac{x^2}{4}$

$y = \sqrt[3]{\left( \dfrac{x}{2} \right)^2}$

$y = (x / 2)^{2/3}$           answer