# Separable DE

• Apr 2nd 2010, 04:37 PM
cdlegendary
Separable DE
http://homework.math.ucsb.edu/webwor...cf1c25d1b1.png

For the equation above, I need to find all the constant solutions. How do I go about doing this? I've tried putting all similar terms on one side and then integrating, but it doesn't seem to work out nicely. Any help is greatly appreciated, thanks in advance!
• Apr 2nd 2010, 04:47 PM
Anonymous1
$\int y' = \int y-y^3$

$\Rightarrow y= \frac{2y^2 - y^4}{4} + C$

$\Rightarrow \frac{y}{4}(y^3-2y+4)=C$

$\Rightarrow \frac{y}{4}(y^3-2y+4)-C=0$

Code:

syms y c solve('(y/4)*(y^3-2*y+4)-c=0',y)   ans =    - 1/6/(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/6)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 1/6/(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/6)/(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/4)*(48*c*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 4*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) + 24*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) + 48*6^(1/2)*(3*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 36*c + 26)^(1/2))^(1/2)   1/6/(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/6)/(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/4)*(48*c*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 4*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) + 24*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) + 48*6^(1/2)*(3*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 36*c + 26)^(1/2))^(1/2) - 1/6/(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/6)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2)   1/6/(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/6)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 1/6/(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/6)/(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/4)*(48*c*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 4*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) + 24*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 48*6^(1/2)*(3*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 36*c + 26)^(1/2))^(1/2)   1/6/(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/6)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) + 1/6/(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/6)/(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/4)*(48*c*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 4*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) + 24*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3)*(12*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(1/3) - 48*c + 9*(8/9*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 32/3*c + 208/27)^(2/3) + 4)^(1/2) - 48*6^(1/2)*(3*3^(1/2)*(64*c^3 + 32*c^2 - 68*c + 25)^(1/2) - 36*c + 26)^(1/2))^(1/2)
umm... lol. Your sure your not given any initial conditions, $y(x)=0?$
• Apr 2nd 2010, 05:08 PM
Krizalid
that's wrong, the ODE actually writes as $\frac{dy}{y-y^{3}}=dx,$
• Apr 2nd 2010, 05:16 PM
Anonymous1
Quote:

Originally Posted by Krizalid
that's wrong, the ODE actually writes as $\frac{dy}{y-y^{3}}=dx,$

Figured it had to be (Wondering). So your pretty good at calc, huh?
• Apr 2nd 2010, 05:45 PM
cdlegendary
uhh yeah there aren't any initial conditions (Headbang)
• Apr 2nd 2010, 05:48 PM
skeeter
Quote:

Originally Posted by cdlegendary
http://homework.math.ucsb.edu/webwor...cf1c25d1b1.png

For the equation above, I need to find all the constant solutions. How do I go about doing this? I've tried putting all similar terms on one side and then integrating, but it doesn't seem to work out nicely. Any help is greatly appreciated, thanks in advance!

if $y = C$ , then $y' = 0$ ... correct?
• Apr 2nd 2010, 06:23 PM
cdlegendary
Quote:

Originally Posted by skeeter
if $y = C$ , then $y' = 0$ ... correct?

ah yes. I think I know where you're going with that
• Apr 2nd 2010, 06:26 PM
cdlegendary
so the solutions are just -1, 0, 1. wow haha. I was definitely thinking too hard about that. thanks!