first-order linear differential equation

A first order linear ODE for the function has the form of

As long as , we can simplify the equation by dividing by :

The above form is called the standard linear form of the first-order linear ODE.

If the coefficients , , (or equivalently, and in the standard form) are constant and don’t depend on , we say that the equation is a constant coefficient ODE.

Subtopics

• first-order linear homogeneous differential equation - If a first order linear ODE is also homogeneous, then it has a simple known solution of

Examples

First-order linear ODEs can model various systems, for example, thermal conduction and particle concentration, radioactive decay, and population growth/decline.

Thermal Conduction

Newton’s law of cooling states that the rate of heat transfer is:

where

• is the temperature
• is the ambient temperature
• is the time
• is the heat transfer coefficient (usually constant)

Initial condition:

Put into standard linear form:

Particle Concentration

The model for particle concentration is mathematically the same to heat conduction:

• is salt concentration inside
• is the salt concentration outside
• is a rate constant