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Consider the problem of calculating the shape of an unknown curve which starts at a given point and satisfies a given differential equation. Here, a differential equation can be thought of as a formula by which the slope of the tangent line to the curve can be computed at any point on the curve, once the position of that point has been calculated.
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable.As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. [1]
The order of the differential equation is the highest order of derivative of the unknown function that appears in the differential equation. For example, an equation containing only first-order derivatives is a first-order differential equation, an equation containing the second-order derivative is a second-order differential equation, and so on.
First-order linear differential equation First-order non-singular perturbation theory First-order partial differential equation , a partial differential equation that involves only first derivatives of the unknown function of n variables
If the process function is Lipschitz-continuous and the associated one-step process has the consistency order , then it also has the convergence order . The Lipschitz continuity of the process function as an additional requirement for stability is generally always fulfilled if the function f {\displaystyle f} from the differential equation ...
First-order means that only the first derivative of y appears in the equation, and higher derivatives are absent. Without loss of generality to higher-order systems, we restrict ourselves to first-order differential equations, because a higher-order ODE can be converted into a larger system of first-order equations by introducing extra variables.
A differential system is a means of studying a system of partial differential equations using geometric ideas such as differential forms and vector fields. For example, the compatibility conditions of an overdetermined system of differential equations can be succinctly stated in terms of differential forms (i.e., for a form to be exact, it ...
In control engineering and system identification, a state-space representation is a mathematical model of a physical system specified as a set of input, output, and variables related by first-order differential equations or difference equations.