Table of contents |

2 A simple mathematical model 3 Improving our model |

The simplest differential equations are ordinary, linear differential equations of the first order with constant coefficients. For example:

- .

Some elaboration is needed since is not in fact a constant, indeed it might not even be integrable. Arguably, one must also assume something about the domains of the functions involved before the equation is fully defined. Are we talking complex functions, or just real, for example? The usual textbook approach is to discuss forming the equations well before considering how to solve them.

Suppose a mass is attached to a spring, which exerts an attractive force on the mass proportional to the extension/compression of the spring and ignore any other forces (gravity, friction etc). We shall write the extension of the spring at a time as . Now, using Newton's second law we can write (using convenient units)

For example, if we suppose at the extension is a unit distance (), and the particle is not moving (). We have

Therefore . (This is an example of simple harmonic motion)

The above model of an oscillating mass on a spring is plausible but not really realistic. For a start, we've invented a perpetual motion machine which violates the second law of thermodynamics. So lets consider adding some friction for realism. Now, experimental scientists will tell us that friction will tend to deccelerate the mass and have magnitude proportional to its velocity (i.e. ). Our new differential equation, expressing the balancing of the acceleration and the forces, is

This is a *damped oscillator*, and the plot of displacement against time would look something like this:

See also Laplace transform, eigenvalues, eigenvectors, vector field, slope field, integration, partial derivative, vector calculus, differential equations of mathematical physics