Differential equation
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A differential equation is a mathematical equation that involves variables like x or y, as well as the rate at which those variables change.
While listening to a weather forecast, the meteorologist might say that the air pressure is 772 and falling. The fact that the air pressure is falling is as important to predicting the weather as the actual pressure itself. This is the kind of situation that might be described by a differential equation.
Many areas of science, engineering, and economics use differential equations. The magnetic field inside an electromagnet depends on the rate at which the electric current passing through it is changing. Force is equal to the rate at which momentum changes. The rate at which the value of the dollar is falling might tell us something about the rate at which the price of gold is rising. In quantum theory, the rate at which the probability of finding an electron around an atom changes can tell us something about how fast energy is entering or leaving the atom. Water and sound waves behave the way they do because they are obeying a differential equation.
If a differential equation involves x, as well as the rate at which x changes, it is called a first order differential equation. If the rate at which the rate of change changes is also involved, it is a second order differential equation. If there are more variables than just x and y, then it is said to be a partial differential equation. Sometimes, something in the world will obey several differential equations at the same time. These are said to be coupled differential equations.
Some differential equations can be solved exactly, and some cannot. Sometimes a computer program can trial-and-error to solve one. Although these relationships may seem overly-complicated to someone who has not studied them before, the people who use differential equations tell us that they would not be able to figure important things out without them. Most physical scientists and engineers (as well as mathematicians) take at least one course in differential equations while in college. Some mathematicians devote their career to investigating differential equations that are difficult to solve.
[change] Uses
Differential equations are used in many fields of science, among others:
- In physics for various forms of movement, or oscillations
- Radioactive decay is calculated using differential equations.
- Isaac Newton's Second law of motion which talks about acceleration
- Newton's Law of Cooling
- The Wave equation
- The heat equation
- The Navier–Stokes equations described the movement of fluids
- The Hamiltonian equations for general mechanics
