1. Applications of 1st Order Differential Equations
Chapter 2
Applications of 1st Order Differential Equations

2. Sec 4.1 – Exponential Growth And Decay
From Calculus:
Natural Growth and Decay
Population growth (bacteria, world population)
Decay (radioactive decay, drug dissipation)
These conform to General Exponential Growth Equation:
Where k is the growth constant (pos or neg)
(Continuously) compounded interest
Cooling & Heating (Newton’s Cooling Law)

3. Sec 4.1 – Exponential Growth And Decay
Recall the basic population equation -
Which has solution function
Where P0=P(0) is the initial population
Our author describes this as a quantity instead of a population, so the formula becomes
If a > 0 then Q increases exponentially. If a < 0, then Q decreases exponentially.

4. Sec 4.1 – Radioactive Decay
The substance is decaying (Q is decreasing), so we know that the constant is < 0. In this case we write
The rate of decrease is usually given as a half-life, the time after which the Q shrinks in half. It is denoted and is such that

5. Example

6. Example

7. Example of different decay rate
Suppose that a population of 10,000 bacteria is hit with a dose of antibiotics, causing the reproduction (birth) rate to stop, and the bacteria to begin to die at the rate of d = .05*P(t)2.
Then P’(t) = -.05*P(t)2
This equation is separable

8. Example of growth AND decay
Suppose that a radioactive substance with decay constant k is being produced at a constant rate a. If the initial quantity is denoted Q0, find a formula for Q(t), and find the quantity as

9. Logistic Equation
In confined populations, birth rate decrease as population grows (food supply, living space, etc.)
In a simple case, instead of a constant, the birth rate is a linear function of P:
If the death rate is constant, the basic equation becomes
Which we can write
Where and
This is the logistic equation

10. Solution of Logistic Equation
The equation
Is separable – So
LHS integrates with partial fractions (or TI)