# What is the differential equation?

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Let #P(t)# represent the number of wolves in a population at time #t# years,

when #t>=0# . The population #P# is increasing at a rate directly proportional to #800-P# , where the constant of proportionality is #k# .

Write a differential equation based on the above information. Solve the equation to find a general expression for the population #P(t)#

Let

when

Write a differential equation based on the above information. Solve the equation to find a general expression for the population

##### 1 Answer

The GS is:

# P = 800-Ae^(- kt) #

#### Explanation:

We are given that

We are asked to write a differential equation based on the above information. Solve the equation to find a general expression for the population

So using the description we have:

# \ \ \ \ \ (dP)/dt prop 800 - P #

# :. (dP)/dt = k(800-P)#

Which is a Separable ODE, so we can *"separate the variables"* to get:

# int \ 1/(800-P) \ dP = int \ k \ dt#

Which we can integrate to get:

# -ln|800-P| = kt+ C #

And we can rearrange:

# ln|800-P| = - kt- C #

# :. |800-P| = e^(- kt- C) #

And noting that

# 800-P = Ae^(- kt) #

So the GS is:

# P = 800-Ae^(- kt) #