Simple Method for Solving y'=y(3-y) ODE

[badtwistoffate]

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 separation of varibles.

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

dy/dx=3y-y^2 then... I am stuck... I tried doing separation 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.

 

[BSMSMSTMSPHD]

$$\frac{dy}{dx} = y(3-y)$$

$$\frac{dy}{y(3-y)} = dx$$

$$\int \frac{dy}{y(3-y)} = \int dx$$

Break up the left-hand side by the method of partial fractions.

 

[wurth_skidder_23]

after partial fractions and integrating both sides:

ln(y) - ln(y-3) = 3 x + Cby logarithmic identities:

ln(y/(y-3)) = 3 x + CTaking the exponential of both sides:

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

Where C1 = e^CSolving for y:

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

 

[BSMSMSTMSPHD]

Well, geez... why don't you just do the whole thing for him? Oh wait, you just did.

Way to encourage the joy of discovery. :uhh:

 

[badtwistoffate]

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?

 

[BSMSMSTMSPHD]

You have the same thing. Just change your - sign into an exponent and multiply both sides by 3.

 

[badtwistoffate]

how do you change a minus sigh into a exponent, I am lost when you said that.

 

[BSMSMSTMSPHD]

Use the property of logarithms: $$r log b = log b^{r}$$

where $$r = -1$$

and $$b = \frac{y-3}{y}$$

 

[HallsofIvy]

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.

 

[badtwistoffate]

It was a brain fart ease up on me