View source for Ordinary differential equation

[[Category:Differential Equations]]

An '''ordinary differential equation (ODE)''' relates a function and its
derivatives. We usually use <math>y</math> to denote the function and
<math>t</math> to denote the variable.

''Ordinary'' means that the equation has one variable, as opposed to
partial differential.

== Example ==

An example of an ODE is the following

<math>
y' = y
</math>

The general solution of the above is

<math>
y(t) = c e^t
</math>

Notably, the solution is ''homogeneous'', meaning that <math>y(t) = 0</math> is
a solution. This will probably be covered later.

= Solving ODE =
An ODE is considered ''solve'' if we can express ''y'' in terms of integrals of explicitly known functions (reduce to quadratures). We don't necessarily need to solve the integral; some integrals are very very hard; MATLAB just does it.

= Classification =
There is ''no general method'' to solve ODEs. We separate them by classes
and solve them individually.

== Linearity ==
An ODE is '''linear''' if all terms are proportional to <math>y, y',
y''. \ldots</math> or are given functions of <math>t</math>.

This
distinction is especially useful since linear combination can be used to
construct solutions. As one see from [[Linear Algebra]], for a linear problem, a general solution can be constructed from ''one specific solution'' and the ''homogeneous solution''.

== Order ==
The '''order''' of an ODE is the order of its highest derivative. Higher order ODEs are generally harder to solve.

== Scalar/System ==
In a '''scalar''', there is only one unknown function <math>y(t)</math>.
In a '''system''', there are several, and you have to solve them
simultaneously (such as predator-prey functions).

== Solvable Classes ==
Here is a list of ODEs we study and solve, from simple to complex:
* [[Linear First Order ODE]]
*[[First Order Scalar ODE|First Order Scalar]]
*[[Autonomous ODE]]
* [[Second order linear ODE]]

* [[Systems of ODEs]]

= Applications =
Since the derivative can be described as the rate of change, and the
function itself is the state, ODEs arises as mathematical models of
systems whose ''rate of change depends on the state of the system''. 

The following are brief descriptions of some applications of ODEs.

# ''Radioactive decay'', where the function is the (large) number of atoms.
#* Atoms decay at an average constant rate <math>r</math>
#* <math>\frac{dN}{dt} = -rN</math>
# ''Object falling under gravity'', where the function is the velocity of the object
#* <math>\frac{dv}{dt} = g - \frac{\gamma v}{m}</math>
Here are some more general questions that are solved by ODEs

* [[Initial Value Problem]]: Use an initial value of a function and an ODE to generate a function.
** [[Equilibrium Solution]]: Find the equilibrium solution of an initial value problem

== Dimensional Analysis ==

The two sides of the equation must match in dimensions (aka. units). This also applies in ODEs.

Consider radioactive decay.

<math>
\begin{cases}
\frac{dN}{dt} = -rN \\
N(0) = N_0
\end{cases}
</math>

The solution comes to

<math>
N(t) = N_0 e^{-rt}
</math>

We use '''time constant''' <math>\tau</math> to get a sense of how fast
it is decaying. Its units is time.

<math>
\tau = \frac{1}{r}
</math>