# How Do You Solve Complex Differential Equations in Population Growth Models?

• AngusYoung93
In summary: P}{P}\,=\,duthen it becomes 2. In summary, the Gompertz equation is a model of population growth with constants a and b. The solution to the differential equation with initial condition P(0) = p(sub(0)) is dP/dt = P(a - b ln(P)). As t approaches infinity, P(t) approaches a/b. The graph of P is concave down. Additionally, the differential equation dP/dt = P(10^-1 - (10^-5)P) models the population of a certain community with initial condition P(0) = 2000. The limit of P(t) as t approaches infinity exists and is
AngusYoung93
Hey, everyone. I've been doing my Calculus II homework, and I've been trying these two problems for the past few hours, but I can't even seem to get started on them. A friend of mine recommended I try here because you guys are awesome.

## Homework Statement

The Gompertz equation
dP/dt = P(a - b ln(P))
where a and b are positive constants, is another model of population growth.
a) Find the solution of this differential equation that satisfies the initial condition: P(0) = p(sub(0))
b) What happens to P(t) as t -> infinity?
c) Determine the concavity of the graph of P

2. The attempt at a solution

dP/dt = P(a - bQ)
where Q = ln(x)​

## Homework Statement

the differential equation
dP/dt = P(10^-1 - (10^-5)P)
models the population of a certain community. Assume P(0) = 2000 and time t is measured in months.
a) Find P(t) and show that lim t -> infinity exists
b) Find the limit

2. The attempt at a solution

I do not even know where to start on this. Any help at all would be nice.

Write it as
$$\frac{d\,P}{P\,(a-b\,\ln P)}=d\,t$$
and set $\ln P=u$

## What is a differential equation?

A differential equation is a mathematical equation that relates a function with its derivatives. It describes how a physical quantity changes over time or space.

## Why are differential equations important?

Differential equations are important because they are used to model many natural phenomena and physical systems, making them crucial in various fields such as physics, engineering, and economics. They help us understand and predict how systems will behave over time.

## How do you solve a differential equation?

There is no one specific method for solving a differential equation, as it depends on the type and complexity of the equation. Some common techniques include separation of variables, substitution, and using integrating factors. In some cases, numerical methods may also be used.

## What are some real-life applications of differential equations?

Differential equations have many real-life applications, such as predicting population growth, modeling the spread of diseases, understanding heat flow and diffusion, and analyzing electrical circuits. They are also used in economics, finance, and biology.

## What skills are needed to work with differential equations?

To work with differential equations, one needs a solid understanding of calculus, algebra, and basic mathematical concepts. Strong problem-solving skills and the ability to think abstractly are also helpful.

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