Modeling Population Growth: Solving for P(t) Using Integration

In summary, we have a population P(t) modeled by the equation dP/dt = 0.0004P(P-150) and we need to find a formula for P(t) at a general time t. To do so, we first swap the equation to dt = 1/0.004P(P-150) dP and then use integration to solve for P(t). We can use partial fractions to rewrite the equation and find A and B, then solve for P(t) using two separate integrals.
  • #1
gtfitzpatrick
379
0

Homework Statement


a population P(t) is modeled by the equation dP/dt = 0.0004P(P-150), Find a formula which gives the population, P(t), at a general time t.

Homework Equations





The Attempt at a Solution



swapping over

dt=1/0.004P(P-150) dP
then i integrate both sides

dt becomes t+c but I'm i'm not sure how to integrate the the dP side, any pointers please?
 
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  • #2
Are you able to integrate [itex]\int\frac{dx}{x}[/itex]?
 
  • #3
First, 1/0.004= 250. You have
[tex]250\int\frac{dP}{P(P-250)}dP[/tex]

Use "partial fractions". Write
[tex]\frac{1}{P(P-250)}= \frac{A}{P}+ \frac{B}{P- 250}[/tex]
Find A and B and do it as two separate integrals.
 

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