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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:
A simple ordinary differential equation
where f(t) is some known function. We may solve this simply by rearranging it (using the chain rule) as
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)
A simple mathematical model
If we look for solutions that have the form , where is a constant, we discover the relationship , and thus must be one of the complex numbers or . Thus, using Euler's theorem we can say that the solution must be of the form:
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
Improving our model
where is our coefficient of friction, and . Again looking for solutions of the form , we find that
This is a quadratic equation which we can solve. If we have complex roots , and the solution (with the above boundary conditions) will look like this:
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