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For an arbitrary system of ODEs, a set of solutions (), …, are said to be linearly-independent if: + … + = is satisfied only for = … = =.A second-order differential equation ¨ = (,, ˙) may be converted into a system of first order linear differential equations by defining = ˙, which gives us the first-order system:
The simplest method for solving a system of linear equations is to repeatedly eliminate variables. This method can be described as follows: In the first equation, solve for one of the variables in terms of the others. Substitute this expression into the remaining equations. This yields a system of equations with one fewer equation and unknown.
The crucial aspect of the method is employment of the "Adomian polynomials" which allow for solution convergence of the nonlinear portion of the equation, without simply linearizing the system. These polynomials mathematically generalize to a Maclaurin series about an arbitrary external parameter; which gives the solution method more ...
The concept can also be applied to more general systems of equations, such as systems of polynomial equations or partial differential equations. In the case of the systems of polynomial equations, it may happen that an overdetermined system has a solution, but that no one equation is a consequence of the others and that, when removing any ...
In mathematics, a differential-algebraic system of equations (DAE) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system. The set of the solutions of such a system is a differential algebraic variety , and corresponds to an ideal in a differential algebra of differential ...
Alabama A&M freshman linebacker Medrick Burnett Jr. died one month after sustaining a head injury in an Oct. 29 game. He was hospitalized for the entire past month.
Ernst Hairer, Syvert Paul Nørsett and Gerhard Wanner, Solving ordinary differential equations I: Nonstiff problems, second edition, Springer Verlag, Berlin, 1993. ISBN 3-540-56670-8. Ernst Hairer and Gerhard Wanner, Solving ordinary differential equations II: Stiff and differential-algebraic problems, second edition, Springer Verlag, Berlin, 1996.
Zillow predicts the US housing market will keep shifting in 2025.. The real-estate firm says the average home value rose by 2.6% annually in October. It says homebuying activity should pick up ...