Find the equilibrium points?

Question

Find the equilibrium points of dy/dt=ay+by^2, a>0, b>0, yo≥0.

Answer 1

You have to show the values of "t" that make the derivative dy/dt = 0. Let's find the explicit solution of the equation. After separation of variables we can rewrite it in the following way:

                                                         dy/y(a+by) = dt                           (1)

The rational expression can be written as

                                         1/y(a + by) = (1/a)[(1/y) - b/(a+by)]             (2)

Integration gives us

                                     (1/a)[lny - ln(a+by)] = ln C + t   (C = const)       (3)

or

                                                    (y/(a+by)) ^1/a = C e ^{t                                (4)}

Raise both sides into "a" power and you will obtain

                                                      y/(a+by)  = C^a e^at = C₁ e ^at               (5)

(C^a is replaced by an arbitrary cxnstant C_1. Taking into account the initial condition y(0) = y₀

we can find that

                                                       y₀ = C₁a/(1-C₁b)                             (6)

Substituiting (6) into (5) and solving for "y" we will obtain the explicit expression for y(t):

                                                    y(t) = a y₀e^at/[a +by₀(1-e^at)]               (7)

If you calculate the derivative dy/dt you will find that it equals zero if  

                                                           a+by₀ = 0   or y₀ = - a/b

Thus, the intial state (t = 0, y₀ = -a/b)   is a state of equilibrium dy/dt = 0.

Despite dy/dt = 0 if y = 0, this solution has to excluded because, as one can see from (7), y = 0 for

t = - ∞ (not a reachable state). In the meantime, for t = ∞  y = -a/b  and dy/dt = 0.

                                                       

Answer 2

The equilibrium points are at the values where dy/dt = 0.

Simply factor the y out of (ay + by^2) to get y(a + by).

ay + by^2 = y(a + by) = 0

a + by = 0 --> a = -by --> y = -a/b

y = 0, -a/b