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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.
For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
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. The idea is that while the curve is initially unknown, its starting point, which we denote by , is known (see Figure 1). Then, from the ...
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.
In numerical analysis, predictor–corrector methods belong to a class of algorithms designed to integrate ordinary differential equations – to find an unknown function that satisfies a given differential equation. All such algorithms proceed in two steps:
Typically, it applies to first-order equations, though in general characteristic curves can also be found for hyperbolic and parabolic partial differential equation. The method is to reduce a partial differential equation (PDE) to a family of ordinary differential equations (ODE) along which the solution can be integrated from some initial data ...
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 ...
Order Equation Application Reference Abel's differential equation of the first kind: 1 = + + + Class of differential equation which may be solved implicitly [1] Abel's differential equation of the second kind: 1