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Consider a linear non-homogeneous ordinary differential equation of the form = + (+) = where () denotes the i-th derivative of , and denotes a function of .. The method of undetermined coefficients provides a straightforward method of obtaining the solution to this ODE when two criteria are met: [2]
Some authors allow any real , [1] [2] whereas others require that not be 0 or 1. [ 3 ] [ 4 ] The equation was first discussed in a work of 1695 by Jacob Bernoulli , after whom it is named. The earliest solution, however, was offered by Gottfried Leibniz , who published his result in the same year and whose method is the one still used today.
Class of differential equation which may be solved implicitly [1] Bernoulli equation: 1 + = Class of differential equation which may be solved exactly [2] Binomial differential equation (′) = (,) Class of differential equation which may sometimes be solved exactly [3]
Ordinary differential equations occur in many scientific disciplines, including physics, chemistry, biology, and economics. [1] In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved.
In mathematics, the binomial differential equation is an ordinary differential equation of the form (′) = (,), where is a natural number and (,) is a polynomial that is analytic in both variables. [ 1 ] [ 2 ]
It was originally known as "HECKE and Manin". After a short while it was renamed SAGE, which stands for ‘’Software of Algebra and Geometry Experimentation’’. Sage 0.1 was released in 2005 and almost a year later Sage 1.0 was released. It already consisted of Pari, GAP, Singular and Maxima with an interface that rivals that of Mathematica.
Numerov's method (also called Cowell's method) is a numerical method to solve ordinary differential equations of second order in which the first-order term does not appear. It is a fourth-order linear multistep method. The method is implicit, but can be made explicit if the differential equation is linear.
In mathematics, the method of characteristics is a technique for solving partial differential equations.Typically, it applies to first-order equations, though in general characteristic curves can also be found for hyperbolic and parabolic partial differential equation.