general solution

Problem 14
$2y \, dx = 3x \, dy$

Solution 14

Problem 15
$my \, dx = nx \, dy$

Solution 15
$my \, dx = nx \, dy$

$m\dfrac{dx}{x} = n\dfrac{dy}{y}$

$m\ln x = n\ln y + \ln c$

$\ln x^m = \ln y^n + \ln c$

$\ln x^m = \ln cy^n$

Problem 12
$\sin x \sin y \, dx + \cos x \cos y \, dy = 0$
 

Solution 12
$\sin x \sin y \, dx + \cos x \cos y \, dy = 0$

$\dfrac{\sin x \sin y \, dx}{\sin y \cos x} + \dfrac{\cos x \cos y \, dy}{\sin y \cos x} = 0$

$\dfrac{\sin x \, dx}{\cos x} + \dfrac{\cos y \, dy}{\sin y} = 0$

$\displaystyle -\int \dfrac{-\sin x \, dx}{\cos x} + \int \dfrac{\cos y \, dy}{\sin y} = 0$

$-\ln (\cos x) + \ln (\sin y) = \ln c$

$\ln \dfrac{\sin y}{\cos x}= \ln c$

$\dfrac{\sin y}{\cos x}= c$