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### Course: AP®︎/College Calculus BC>Unit 7

Lesson 9: Logistic models with differential equations

# Growth models: introduction

Population should grow proportionally to its size, but it can't keep growing forever! Learn more about this problem, posed by Malthus, and embark on a journey towards its mathematical solution.

## Want to join the conversation?

• How is it allowed to multiply by dt as dt is a number. dN/dt is not a quotient so that we could multiply it and cancel out the dt, right? What does this multiplication with dt equate to, mathematically?
• In reality you are integrating both sides with respect to t. Let us say your differential equation is dN/dt = f(t)/h(N(t)). Thus h(N(t)) dN/dt = f(t). Integrating with respect to t gives
\int h(N(t)) dN/dt dt = \int f(t) dt

The left integral can be integrated by using substitution u = h(N(t)), du = h'(N(t)) dN/dt dt.
\int h(u) du = \int f(t) dt
Then you can integrate.
• With the absolute value of N, i know it works logically getting rid of it since the population will always be positive, but can't you work it out mathematically once you have

|N| = e^(rt+c)

Since the right side of the equation will always be positive?
• Isnt "r" the rate of growth?
• no, it's a constant. it's related to the rate of growth.
• Sorry, I wonder that what is "r" ?
• I've noticed that most problems use e. How is e derived?
• The constant e is defined as the limit of (1 + 1/n)^n as n approaches infinity. It can also be calculated from the infinite series (1 / n!) from n=0 to infinity.
• at rough you state N must be greater than 0. Population if over harvested can be zero can it not?
• i think he stated N to be greater than 0 because the function ln(x) goes nuts (towards negative infinity) on x=0 which screws the whole equation.
• If N(t)=No*e^rt is a solution to the original differential equation, shouldn't you be able to take the derivative of it with respect to t and get rN? Taking dN/dt of the solution I get: dN/dt=Nor*e^rt
• Yes, and indeed:
Differential equation is: dN/dt = rN
You correctly derived the left-hand side: Nor*et
The right-hand side = rN = r*Noe^rt
LHS = RHS
OK?
(1 vote)
• at , what happens to dN? I understand that 1/N goes to ln N. But why does dt go to just t while dN just disappear?
(1 vote)
• Once you take the integral of something, the dN and the dT will go away. That's just the way it is. For example, if you take the integral of x^2 dx, the answer would be x^3/3 and the dx goes away.