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Ordinary Differential Equation (ODE) - Coggle Diagram
Ordinary Differential Equation
(ODE)
Definition
An ordinary differential equation (ODE) is an equation involving: X, Y, Y', Y''...
Solution
A solution of an ODE is a function y(x) whose derivatives satisfy the equation. It is not guaranteed that such a function exists, and if it does, it is usually not unique.
Linearity
As for linearity: an ordinary differential equation of order n can be seen as a function
We say that an ODE is linear if F is linear in Y, Y'(x), Y''(x)...
Explicit ODE / Implicit ODE
When an N-order ODE is of the form
F(X, Y, Y', Y''..., Y^n) = 0
we say that it is implicit.
When an N-order ODE is of the form
F(X, Y, Y', Y''..., Y^(n-1)) = Y^n
we say that it is explicit.
Autonomous ODE / Homogeneous ODE
A ODE is autonomous if it does not explicitly depend on x, and homogeneous if all the terms of the differential equation depend exclusively on x.
Resolution Methods
Exact differential equation
Order Reduction
Parameter variation method
Coefficients to be determined
Separable Equations
First order homogeneous equations
Integrating Factor Method
Reducible to homogeneous
Applications
Differential equations are used very often to describe processes in which a change in a measure or dimension is caused by the process itself.Two well-known equations where ODE are used are in the half-life law equation and in the harmonic unchanged oscillator.
