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[✓] Find the critical solutions?

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The question states, "Given that dP/dx=3P-2P^2 for the population P of a certain species at time t, find the critical solutions (equilibrium solutions)" I'm not exactly sure how to do that. I've tried using DSolveValue and Solve but they give two different answers so I'm not sure which one is right or if either are right. I'm new to mathematica so I don't know what to use

   In[3]:= (*2.1*)
DSolveValue[P'[x] == 3*P[x] - 2*P[x]^2, P, x]

Out[3]= Function[{x}, (3 E^(3 x))/(2 E^(3 x) + E^(3 C[1]))]
 In[6]:= Clear[P, x]
    Solve[P'[x] == 3*P[x] - 2*P[x]^2, P[x]]


    Out[7]= {{P[x] -> 1/4 (3 - Sqrt[9 - 8 Derivative[1][P][x]])}, {P[x] ->
    1/4 (3 + Sqrt[9 - 8 Derivative[1][P][x]])}}
POSTED BY: Brendan Isaac
Answer
2 months ago

Since you haven't supplied any initial condition for your DE there will be an unknown constant in the general solution

f = FullSimplify[P[x]/.DSolve[P'[x] == 3 P[x] - 2 P[x]^2, P[x], x][[1]]]

which gives you

3/(2 + E^(-3 x + 3 C[1]))

You can peek at a plot for a handful of values picked for the constant C[1], but C[1] doesn't have to be an integer, and see the behavior

Plot[{(f/.C[1]-> -2), (f/.C[1]-> -1), (f/.C[1]-> 0), (f/.C[1]-> 1), (f/.C[1] -> 2)}, {x, -10, 10}]

plot

POSTED BY: Bill Simpson
Answer
2 months ago

The equilibrium solutions are those with P'[x] is identically zero. You get them with Solve[3P-2P^2,P].

POSTED BY: Gianluca Gorni
Answer
2 months ago

The other solution is,

f = FullSimplify[P[x] /. DSolve[P'[x] == 3 P[x] - 2 P[x]^2, P[x], x][[1]]]

and

f /. x -> Infinity

It brings us the stable value

3/2

for all initial conditions.

POSTED BY: Yoshino Takashi
Answer
2 months ago

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