Ordinary differential equation: Difference between revisions

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a solution. This will probably be covered later.
a solution. This will probably be covered later.


= Initial Value Problem =
= Classification =
To get a unique solution, we need to apply additional conditions, such
as specifying a particular value


<math>
An ODE is '''linear''' if all terms are proportional to <math>y, y',
\begin{cases}
y''. \ldots</math> or are given functions of <math>t</math>. This
y' = y \\
distinction is especially useful since linear combination can be used to
y(0) = y_0
construct solutions.
\end{cases}
 
</math>
The '''order''' of an ODE is the order of its highest derivative.


This is called an ''initial value problem'', in which a function is
In a '''scalar''', there is only one unknown function <math>y(t)</math>.
generated from an initial value with another equation.
In a '''system''', there are several, and you have to solve them
simultaneously.


== Usage ==
Here is a list of ODEs we study, from simple to complex:
* [[Linear First Order ODE]]


= Applications =
Since the derivative can be described as the rate of change, and the
Since the derivative can be described as the rate of change, and the
function itself is the state, ODEs arises as mathematical models of
function itself is the state, ODEs arises as mathematical models of
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# ''Object falling under gravity'', where the function is the velocity of the object
# ''Object falling under gravity'', where the function is the velocity of the object
#* <math>\frac{dv}{dt} = g - \frac{\gamma v}{m}</math>
#* <math>\frac{dv}{dt} = g - \frac{\gamma v}{m}</math>
Here are some more general questions that are solved by ODEs


== Dimensions/Units ==
* [[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


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


Consider radioactive decay.
Consider radioactive decay.
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\tau = \frac{1}{r}
\tau = \frac{1}{r}
</math>
</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 21:26, 15 April 2024


An ordinary differential equation (ODE) relates a function and its derivatives. We usually use Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} to denote the function and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} to denote the variable.

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

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

Example

An example of an ODE is the following

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y' = y }

The general solution of the above is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t) = c e^t }

Notably, the solution is homogeneous, meaning that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t) = 0} is a solution. This will probably be covered later.

Classification

An ODE is linear if all terms are proportional to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y, y', y''. \ldots} or are given functions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} . 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t)} . 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:

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.

  1. Radioactive decay, where the function is the (large) number of atoms.
    • Atoms decay at an average constant rate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r}
    • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dN}{dt} = -rN}
  2. Object falling under gravity, where the function is the velocity of the object
    • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dv}{dt} = g - \frac{\gamma v}{m}}

Here are some more general questions that are solved by ODEs

Dimensional Analysis

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

Consider radioactive decay.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} \frac{dN}{dt} = -rN \\ N(0) = N_0 \end{cases} }

The solution comes to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N(t) = N_0 e^{-rt} }

We use time constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau} to get a sense of how fast it is decaying. Its units is time.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau = \frac{1}{r} }

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