How Do You Find Inflection Points Using Differential Equations?

[ssb]

Homework Statement

For many differential equations, the easiest way to find inflection points is to use the differential equation rather than the solution itself. To do this, we can compute y'' by differentiating y', remembering to use the chain rule wherever y occurs. Next, we can substitute for y' by using the differential equation and setting y' = 0. Then we can solve for y to find the inflection points. (Keep in mind here that solving for y can also produce some equilibrium solutions, which may not be inflection points!)

Use the technique described above to find the inflection point for the solutions of the differential equation


y'=r(1-\frac{y}{L})y

your answer may contain L and r



y = ?

The Attempt at a Solution

I differentiated the given equation and set it equal to zero, then I solved it for y. My answer was Lr/4 but this is wrong according to webworks.

The equation I got when I differentiated y'=r(1-\frac{y}{L})y was y'' = r-((4y)/L)

i know the answer is L/2 but I don't know how to get there.

 

[skeeter]

y' = ry(1 - y/L)
distribute ...
y' = ry - ry²/L
differentiate ...
y" = r - 2ry/L
set y" = 0 ...
r - 2ry/L = 0
r(1 - 2y/L) = 0
1 = 2y/L
y = L/2