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#REDIRECT [[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.
 
There is ''no general solution'' to ODEs. We separate them by classes
and solve them individually.
 
== 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>0</math> is
a solution. This will probably be covered later.
 
To get a unique solution, we need to apply additional conditions, such
as specifying a particular value
 
<math>
\begin{cases}
y' = y \\
y(0) = y_0
\end{cases}
</math>
 
This is called an ''initial value problem'', in which a function is
generated from an initial value with another equation.
 
== Usage ==
 
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>
 
== Dimensions/Units ==
 
The two sides of the equation must match in dimensions (aka. units).
 
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>
 
== Equilibrium Solution ==
 
Consider an object falling under gravity
 
<math>
\begin{cases}
\frac{dv}{dt} = g - \lambda v \\
v(0) = v_0
\end{cases}
</math>
 
We sometimes want the '''equilibrium solution'''
 
<math>
v(t) = v_*
</math>
 
<math>
\frac{dv}{dt} = 0 = g - \lambda v_*
</math>
 
Doing some math, we can eventually get
 
<math>
v(t) = v_* + (v_0 - v_*) e^{-\lambda t}
</math>
 
= Classification =
 
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.
 
The '''order''' of an ODE is the order of its highest derivative.
 
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.
 
Here is a list of ODEs we study, from simple to complex:
* [[Linear First Order ODE]]

Revision as of 21:13, 8 April 2024


An ordinary differential equation (ODE) relates a function and its derivatives. We usually use to denote the function and to denote the variable.

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

There is no general solution to ODEs. We separate them by classes and solve them individually.

Example

An example of an ODE is the following

The general solution of the above is

Notably, the solution is homogeneous, meaning that is a solution. This will probably be covered later.

To get a unique solution, we need to apply additional conditions, such as specifying a particular value

This is called an initial value problem, in which a function is generated from an initial value with another equation.

Usage

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.

  1. Radioactive decay, where the function is the (large) number of atoms.
    • Atoms decay at an average constant rate
  2. Object falling under gravity, where the function is the velocity of the object

Dimensions/Units

The two sides of the equation must match in dimensions (aka. units).

Consider radioactive decay.

The solution comes to

We use time constant to get a sense of how fast it is decaying. Its units is time.

Equilibrium Solution

Consider an object falling under gravity

We sometimes want the equilibrium solution

Doing some math, we can eventually get

Classification

An ODE is linear if all terms are proportional to or are given functions of . This distinction is especially useful since linear combination can be used to construct solutions.

The order of an ODE is the order of its highest derivative.

In a scalar, there is only one unknown function . In a system, there are several, and you have to solve them simultaneously.

Here is a list of ODEs we study, from simple to complex: