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

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)

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.

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

Example

Example

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

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

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

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