separation of variables | MATHalino reviewer about separation of variables

Problem 11
$(1 - x)y' = y^2$

Solution 11

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$(1 - x)y' = y^2$

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

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

Solution 09

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$(2a^2 - r^2) \, dr = r^3 \sin \theta \, d\theta$

$\dfrac{(2a^2 - r^2) \, dr}{r^3} = \sin \theta \, d\theta$

Problem 07
$y' = x \exp (y - x^2)$, when $x = 0$, $y = 0$.

Solution 07

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$y' = x \exp (y - x^2)$

$\dfrac{dy}{dx} = x e^{y - x^2}$

Problem 05
$2y \, dx = 3x \, dy$, when $x = -2$, $y = 1$.

Solution 05
From Solution 04,
$\dfrac{x^2}{y^3} = c$

Problem 03
$xy \, y' = 1 + y^2$, when $x = 2$, $y = 3$.

Solution 03
$xy \, y' = 1 + y^2$

$xy \dfrac{dy}{dx} = 1 + y^2$

Problem 02
$2xy \, y' = 1 + y^2$, when $x = 2$, $y = 3$.

Solution 2
$2xy \, y' = 1 + y^2$

$2xy \dfrac{dy}{dx} = 1 + y^2$

Problem 04
$2y \, dx = 3x \, dy$, when $x = 2$, $y = 1$.

Solution 04
$2y \, dx = 3x \, dy$

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