Finding the limit as x approaches infinity given graph

Question

Hi,

The question reads:

The differential equation dy/dx = k(M-y)y shown has a particular solution with the initial condition f(2) = 2.

Let y = h(x) be a particular solution to the given differential equation f(x, y) with h(4) = 10. Find lim (x--> infinity) [(6-h(x))/(f(x)-6)].

*Unfortunately, I was unable to attach the graph -- it is a slope field graph with a carrying capacity of 6 (horizontal asymptote at 6)

I am unsure of the wording of the question and I will appreciate any help. Thank you!

Answer 1

This DE is a "variables separable" equation:

dy/[(M-y)y] = k dx

(1/M){[1/(M-y)]+(1/y)} = 1/[(M-y)y]

Integrate this partial fraction decomposition to

(1/M)[ln(M-y) + ln y] = kx + C

This is the general solution. You should be able to answer the questions with this information.