# Differental Equation homework (w/ calc 2 intergrals)

• mandymandy
In summary, to solve for P(t) in a population with a carrying capacity of N, assuming a small population and a proportional change rate, we can use the equation dP/dT = KP(N-P) and the technique of partial fractions. This involves integrating 1/[P(N-P)] and using the hint given by the teacher, 1/[P(N-P)] = A/P + B/(N-P).
mandymandy

## Homework Statement

Carrying capacity N of a population

Assume: small population, dp/dt is proportional to P

if 0<P<N then p(t) increases
if P>N then P(t) decreases
equilibrium means dp/dt = 0

dP/dT = KP(N-P)

Solve for P(t)

## The Attempt at a Solution

∫dP/P(N-P) = ∫kdt ?

My teacher said we'd be using ln(...) and 1/[P(N-P)] = A/P + B/(N-P)

mandymandy said:

## Homework Statement

Carrying capacity N of a population

Assume: small population, dp/dt is proportional to P

if 0<P<N then p(t) increases
if P>N then P(t) decreases
equilibrium means dp/dt = 0

dP/dT = KP(N-P)

Solve for P(t)

## The Attempt at a Solution

∫dP/P(N-P) = ∫kdt ?

My teacher said we'd be using ln(...) and 1/[P(N-P)] = A/P + B/(N-P)

You are on the right track. Perhaps you need to review partial fractions to work the integral$$\int \frac{1}{P(N-P)}\, dP$$Your teacher gave you a pretty good hint there.

I can solve it from here I don't know why I was so confused. :) thanks

## 1. What are the basic concepts in differential equations?

The basic concepts in differential equations include the order and degree of the equation, the types of equations (ordinary, partial, linear, nonlinear), and the methods of solving (analytical, numerical).

## 2. How do I determine the order and degree of a differential equation?

The order of a differential equation is the highest derivative present in the equation, while the degree is the highest power of the derivative. For example, in the equation y'' + 3xy' = 2x, the order is 2 and the degree is 1.

## 3. What is the difference between ordinary and partial differential equations?

Ordinary differential equations involve a single independent variable and its derivatives, while partial differential equations involve multiple independent variables and their partial derivatives. ODEs are often used to model systems with one variable, while PDEs are used for systems with multiple variables.

## 4. How do I solve differential equations using integration?

The method of separation of variables is often used to solve differential equations with integration. This involves isolating the variables on opposite sides of the equation and then integrating both sides. Other methods include using integrating factors and substitution.

## 5. What are some real-world applications of differential equations?

Differential equations are used to model various natural phenomena, such as population growth, heat transfer, fluid dynamics, and electrical circuits. They are also used in fields such as engineering, physics, biology, and economics to study and predict complex systems.

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