Ordinary differential equation: Difference between revisions
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''Ordinary'' means that the equation has one variable, as opposed to | ''Ordinary'' means that the equation has one variable, as opposed to | ||
partial differential. | partial differential. | ||
== Example == | == Example == | ||
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Notably, the solution is ''homogeneous'', meaning that <math>y(t) = 0</math> is | Notably, the solution is ''homogeneous'', meaning that <math>y(t) = 0</math> is | ||
a solution. This will probably be covered later. | 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 = | = Classification = | ||
There is ''no general method'' to solve ODEs. We separate them by classes | |||
and solve them individually. | |||
== Linearity == | == Linearity == | ||
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*[[First Order Scalar ODE|First Order Scalar]] | *[[First Order Scalar ODE|First Order Scalar]] | ||
*[[Autonomous ODE]] | *[[Autonomous ODE]] | ||
* [[Second order linear ODE]] | |||
* [[Systems of ODEs]] | |||
= Applications = | = Applications = |
Latest revision as of 06:51, 10 June 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.
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.
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 or are given functions of .
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 . 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:
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
- Object falling under gravity, where the function is the velocity of the object
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.
The solution comes to
We use time constant to get a sense of how fast it is decaying. Its units is time.