Variables Separable

Problem 18
$ye^{2x} \, dx = (4 + e^{2x}) \, dy$
 

Solution 18
$ye^{2x} \, dx = (4 + e^{2x}) \, dy$

$\dfrac{ye^{2x} \, dx}{y(4 + e^{2x})} = \dfrac{(4 + e^{2x}) \, dy}{y(4 + e^{2x})}$

$\dfrac{e^{2x} \, dx}{4 + e^{2x}} = \dfrac{dy}{y}$

$\displaystyle \dfrac{1}{2} \int \dfrac{e^{2x} (2 \, dx)}{4 + e^{2x}} = \int \dfrac{dy}{y}$

$\frac{1}{2} \ln (4 + e^{2x}) = \ln y + \ln c$

$\frac{1}{2} \ln (4 + e^{2x}) = \ln cy$

$\ln (4 + e^{2x}) = 2\ln cy$

$\ln (4 + e^{2x}) = \ln (cy)^2$

$\ln (4 + e^{2x}) = \ln c^2y^2$

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

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

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

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

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

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

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

Problem 08
$xy^2 \, dx + e^x \, dy = 0$,   when   $x \to \infty$,   $y \to \frac{1}{2}$.
 

Solution 08
$xy^2 \, dx + e^x \, dy = 0$

$\dfrac{xy^2 \, dx}{y^2 e^x} + \dfrac{e^x \, dy}{y^2 e^x} = 0$

$\dfrac{x \, dx}{e^x} + \dfrac{dy}{y^2} = 0$

$\displaystyle \int xe^{-x} \, dx + \int y^{-2} \, dy = 0$
 

For   $\displaystyle \int xe^{-x} \, dx$
Let
$u = x$,   $du = dx$

$dv = \int e^{-x} \, dx$,   $v = -e^{-x}$