The slop field for the equation dP/dt = 0.0333333 P(30 - P), for P greaterthanor

Question

The slop field for the equation dP/dt = 0.0333333 P(30 - P), for P greaterthanorequalto 0, is shown below. On a print out of this slope field, sketch the solutions that pass through (0, 0); (3, 12); (12, 3); (-14.5, 3): (-6, 36): and (-6, 30) For which positive values of Pare the solutions increasing? increasing for: (Give your answer as an interval or list of intervals, e.g., if P is increasing between 1 and 5 and between 7 and infinity, enter (1, 5), (7, Inf) For what positive values of Pare the solutions decreasing? Decreasing for: Again, give your answer as an interval or list of intervals, e.g., if P is decreasing between 1 and 5 and between 7 and infinity, enter (1, 5), (7, Inf) What is the equation of the Solution to this differential equation that passes through (0, 0)? P = If the solution passes through a value of P > 0, what is the limiting value of P as t gets large? P rightarrow

Explanation / Answer

A) P(t) is increasing when dP/dt > 0
0.0333333P(30P) > 0
P(30P) > 0
0 < P < 30
Solution: (0, 30)

B) P(t) is decreasing when dP/dt < 0
P < 0 or P > 30
Since we are looking for positive values of P, then interval is (30, )

C) P = 0 seems correct

D) Limiting value of P occurs when dP/dt = 0
0.0333333P(30P) = 0
P = 0, or P = 30
Since we are looking at solutions through a value of P > 30, then
limiting value as t gets large is P = 30


Notice how answer to D) matches answers to A) and B)
When P < 30, then P increases
When P > 30, then P decreases
So we can see that P always goes toward 30
Also, we could have solved differential equation to get
P(t) = 30 e^t / (C + e^t)
And then find limit at t approaches infinity:
lim[t] 30 e^t / (C + e^t)
= lim[t] 30 / (C e^(-t) + 1)
= 30 / (0 + 1)
= 30

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