Models Using Differential Equations
Differential equations can be used to model a variety of physical systems.
Learning Objective

Give examples of systems that can be modeled with differential equations
Key Points
 Many systems can be well understood through differential equations.
 Mathematical models of differential equations can be used to solve problems and generate models.
 An example of such a model is the differential equation governing radioactive decay.
Terms

decay
To change by undergoing fission, by emitting radiation, or by capturing or losing one or more electrons.

differential equation
an equation involving the derivatives of a function
Full Text
Differential equations are very important in the mathematical modeling of physical systems.
Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same secondorder partialdifferential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat is governed by another secondorder partial differential equation, the heat equation .
Visual Model of Heat Transfer
Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution.
A good example of a physical system modeled with differential equations is radioactive decay in physics.
Over time, radioactive elements decay. The halflife,
We can combine these quantities in a differential equation to determine the activity of the substance. For a number of radioactive particles
a firstorder differential equation.
Key Term Reference
 average
 Appears in these related concepts: Mean: The Average, Average Value of a Function, and Averages
 differential
 Appears in these related concepts: Triple Integrals in Cylindrical Coordinates, Thermal Stresses, and Compensation Differentials
 inverse
 Appears in these related concepts: Inverse Functions, The Law of Universal Gravitation, and Hyperbolic Functions
 mathematical model
 Appears in these related concepts: Essential Functions for Mathematical Modeling, Applications and Mathematical Models, and Linear Mathematical Models
 mean
 Appears in these related concepts: Mean, Variance, and Standard Deviation of the Binomial Distribution, The Mean Value Theorem, Rolle's Theorem, and Monotonicity, and Understanding Statistics
Sources
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