Examples of differential equations

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Table of contents

1 A simple ordinary differential equation
2 Non-seperable first order linear ordinary differential equations
3 A simple mathematical model
4 Improving our model
5 A simple exact equation
6 See also

A simple ordinary differential equation

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

where f(t) is some known function. We may solve this simply by rearranging it (using the chain rule) as

where

Integrating this, we have

where A is an arbitrary constant. (We can easily check this is a solution)

Some elaboration is needed since f(t) 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.

Non-seperable first order linear ordinary differential equations

Some first order linear ODEs(ordinary differential equations) are not seperable like in the above example. In order to solve non-seperable first order linear ODEs one must use what is known as an integrating factor. This technique will be shown below.

Consider first order linear ODEs of the general form:

This equation can not be separated and integrated until you find the integrating factor , such that:

After integrating with respect to and finding , you multiply both sides of the equation by to get:

This simplifies to:

Using the chain rule we get:

Since and are both functions of , their product will also be a function of , which we will call . This makes our equation look like this:

Integrating both sides we get:

Finally, to solve for we divide both sides by :

A simple mathematical model

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)

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:

To fix the unknown constants and , we need initial conditions, i.e. to specify the state of the system at a given time (usually taken to be ).

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

and so .

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

Improving our model

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

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:

(We can show that )

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

which does resemble how we'd expect a vibrating spring to behave as friction removed the energy from the system.

A simple exact equation

An exact differential equation is a first-order ordinary differential equation of implicit form

such that

This equation has the solution

where

u and v being dummy variables; x₀ and y₀ being initial-value constants.