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 17
$\dfrac{dV}{dP} = -\dfrac{V}{P}$
 

Solution 17

Problem 16
$y' = xy^2$

Solution 16

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 13
$xy^3 \, dx + e^{x^2} \, dy = 0$
 

Solution 13

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 11
$(1 - x)y' = y^2$
 

Solution 11

Problem 10
$v (dv / dx) = g$,   when   $x = x_o$,   $v = v_o$.
 

Solution 10

Problem 09
$(2a^2 - r^2) \, dr = r^3 \sin \theta \, d\theta$,   when   $\theta = 0$,   $r = a$.
 

Solution 09