particular solution

Problem 10 | Separation of Variables

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

Solution 10

Problem 09 | Separation of Variables

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

Solution 09

Problem 08 | Separation of Variables

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
$u = x$,   $du = dx$

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

Problem 07 | Separation of Variables

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

Solution 07

Problem 05 | Separation of Variables

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

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

Problem 06 | Separation of Variables

Problem 06
$2y \, dx = 3x \, dy$,   when   $x = 2$,   $y = -1$.
 

Solution 06
From Solution 04,
$\dfrac{x^2}{y^3} = c$
 

Problem 03 | Separation of Variables

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 | Separation of Variables

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 | Separation of Variables

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

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

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

Problem 01 | Separation of Variables

Problem 01
$\dfrac{dr}{dt} = -4rt$,   when   $t = 0$,   $r = r_o$
 

Solution 01
$\dfrac{dr}{dt} = -4rt$

$\dfrac{dr}{r} = -4t\,dt$
 

 
 
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