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The following differential equation is separable as it is of the form dP/dt = g(P)h(t).

dP/dt = P − P2

Find the following antiderivatives. (Use C for the constant of integration. Remember to use absolute values where appropriate.)

∫ dP /g(P) = _________
∫ h(t) dt =__________

Solve the given differential equation by separation of variables.

Respuesta :

Answer:

[tex]P(1-P) = Ae^t[/tex]

Step-by-step explanation:

given differential equation is

[tex]\frac{dP}{dt} =P-P^2\\\frac{dP}{dt} = g(P) h(t)\\g(P) = P-P^2\\h(t) = 1[/tex]

So this is separable

We separate as

[tex]\frac{dP}{P-P^2} =1dt\\\frac{dP}{P(1-P)} =1dt[/tex]

Resolve into partial fractions to solve this

[tex]dP(\frac{1}{P}+ \frac{1}{1-P} ) = 1dt\\ln P(1-P) = t+C\\P(1-P) = Ae^t[/tex]

So solution is[tex]P(1-P) = Ae^t[/tex]

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