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Help solving y'=y(3-y)

  • #1
just started ordinary differential equation class, and i have to find the general soln of the eqn: y'=y(3-y)... just started so i only know of the method of seperation of varibles.

Thought of multiplying the y through so it becomes:
y'=3y-y^2 then do..

dy/dx=3y-y^2 then... im stuck... I tried doing seperation of varibles by

Dividing 3y-y^2 to the other side so you get:

dy/3y-y^2=0? But that doesn't seem right.
 
Last edited:

Answers and Replies

  • #2
131
0
[tex] \frac{dy}{dx} = y(3-y) [/tex]

[tex] \frac{dy}{y(3-y)} = dx [/tex]

[tex] \int \frac{dy}{y(3-y)} = \int dx [/tex]

Break up the left-hand side by the method of partial fractions.
 
  • #3
after partial fractions and integrating both sides:

ln(y) - ln(y-3) = 3 x + C


by logarithmic identities:

ln(y/(y-3)) = 3 x + C


Taking the exponential of both sides:

y/(y-3) = C1 e^(3 x)

Where C1 = e^C


Solving for y:

y = 3 C1 e^(3 x)/(C1 e^(3 x)-1)
 
  • #4
131
0
Well, geez... why don't ya just do the whole thing for him? Oh wait, you just did.

Way to encourage the joy of discovery. :uhh:
 
  • #5
wurth_skidder_23 said:
after partial fractions and integrating both sides:

ln(y) - ln(y-3) = 3 x + C


by logarithmic identities:

ln(y/(y-3)) = 3 x + C


Taking the exponential of both sides:

y/(y-3) = C1 e^(3 x)

Where C1 = e^C


Solving for y:

y = 3 C1 e^(3 x)/(C1 e^(3 x)-1)

After my partial fraction decomposition I got:

-ln((y-3)/y)/3=x+c?
 
  • #6
131
0
You have the same thing. Just change your - sign into an exponent and multiply both sides by 3.
 
  • #7
how do you change a minus sigh into a exponent, im lost when you said that.
 
  • #8
131
0
Use the property of logarithms: [tex] r log b = log b^{r} [/tex]

where [tex] r = -1 [/tex]

and [tex] b = \frac{y-3}{y} [/tex]
 
  • #9
HallsofIvy
Science Advisor
Homework Helper
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I'm afraid you are going to find differential equations extremely difficult if you cannot do basic algebra!
Starting from dy/dx=3y-y2 and dividing both sides by 3y- y2, you do NOT get dy/(3y- y2)= 0 any more than dividing both sides of xy= 3 by 3 would give xy/3= 0.
 
  • #10
It was a brain fart ease up on me
 

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