A differential equation is an equation involving a function and one or more of its derivatives. A solution is a function that satisfies the differential equation when and its derivatives are substituted into the equation.
The order of a differential equation is the highest order of any derivative of the unknown function that appears in the equation.
A differential equation coupled with an initial value is called an initial-value problem. To solve an initial-value problem, first find the general solution to the differential equation, then determine the value of the constant. Initial-value problems have many applications in science and engineering.
When studying population functions, different assumptions—such as exponential growth, logistic growth, or threshold population—lead to different rates of growth.
The logistic differential equation incorporates the concept of a carrying capacity. This value is a limiting value on the population for any given environment.
The logistic differential equation can be solved for any positive growth rate, initial population, and carrying capacity.
Any first-order linear differential equation can be written in the form
We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may or may not include an initial value.
Applications of first-order linear differential equations include determining motion of a rising or falling object with air resistance and finding current in an electrical circuit.