Ordinary Differential Equation (ODE)

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An Ordinary Differential Equation (ODE) is a differential equation that contains one independent variable and its derivatives.



References

2015

  • (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/ordinary_differential_equation#System_of_ODEs Retrieved:2015-2-20.
    • A number of coupled differential equations form a system of equations. If y is a vector whose elements are functions; y(x) = [y1(x), y2(x),..., ym(x)], and F is a vector valued function of y and its derivatives, then : [math] \mathbf{y}^{(n)} = \mathbf{F}\left(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n-1)} \right) [/math] is an explicit system of ordinary differential equations of order or dimension m. In column vector form: : [math] \begin{pmatrix} y_1^{(n)} \\ y_2^{(n)} \\ \vdots \\ y_m^{(n)} \end{pmatrix} = \begin{pmatrix} F_1 \left (x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n-1)} \right ) \\ F_2 \left (x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n-1)} \right ) \\ \vdots \\ F_m \left (x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n-1)} \right) \\ \end{pmatrix} [/math] These are not necessarily linear. The implicit analogue is: : [math] \mathbf{F} \left(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n)} \right) = \boldsymbol{0} [/math] where 0 = (0, 0,... 0) is the zero vector. In matrix form : [math] \begin{pmatrix} F_1(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n)}) \\ F_2(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n)}) \\ \vdots \\ F_m(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\cdots \mathbf{y}^{(n)}) \\ \end{pmatrix}=\begin{pmatrix} 0\\ 0\\ \vdots\\ 0\\ \end{pmatrix} [/math] For a system of the form [math] \mathbf{F} \left(x,\mathbf{y},\mathbf{y}'\right) = \boldsymbol{0} [/math] , some sources also require that the Jacobian matrix [math] \frac{\partial\mathbf{F}(x,\mathbf{u},\mathbf{v})}{\partial \mathbf{v}} [/math] be non-singular in order to call this an implicit ODE [system]; an implicit ODE system satisfying this Jacobian non-singularity condition can be transformed into an explicit ODE system. In the same sources, implicit ODE systems with a singular Jacobian are termed differential algebraic equations (DAEs). This distinction is not merely one of terminology; DAEs have fundamentally different characteristics and are generally more involved to solve that (nonsigular) ODE systems. Presumably for additional derivatives, the Hessian matrix and so forth are also assumed non-singular according to this scheme, although note that any ODE of order greater than one can [and usually is] rewritten as system of ODEs of first order, which makes the Jacobian singularity criterion sufficient for this taxonomy to be comprehensive at all orders.


2012

  • http://en.wikipedia.org/wiki/Ordinary_differential_equation
    • QUOTE: In mathematics, an ordinary differential equation (abbreviated ODE) is an equation containing a function of one independent variable and its derivatives. There are many general forms an ODE can take, and these are classified in practice (see below).[1][2] The derivatives are ordinary because partial derivatives only apply to functions of many independent variables (see Partial differential equation).

      The subject of ODEs is a sophisticated one (more so with PDEs), primarily due to the various forms the ODE can take and how they can be integrated. Linear differential equations are ones with solutions that can be added and multiplied by coefficients, and the theory of linear differential equations is well-defined and understood, and exact closed form solutions can be obtained. By contrast, ODEs which do not have additive solutions are non-linear, and finding the solutions is much more sophisticated because it is rarely possible to represent them by elementary functions in closed form - rather the exact (or "analytic") solutions are in series or integral form. Frequently graphical and numerical methods are used to generate solutions, by hand or on computer (only approximately, but possible to do very accurately depending on the specific method used), because in this way the properties of the solutions without solving them can still yield very useful information, which may be all that is needed.