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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]
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.
The conjugate gradient method can also be used to solve unconstrained optimization problems such as energy minimization. It is commonly attributed to Magnus Hestenes and Eduard Stiefel, [1] [2] who programmed it on the Z4, [3] and extensively researched it. [4] [5] The biconjugate gradient method provides a generalization to non-symmetric matrices.
Wolfram Research: 14.2.0 (January 23, 2025; 29 days ago (7] Regularly: Proprietary: Linux, Mac OS X, Windows, Raspbian, Online service. MATLAB Partial Differential Equation Toolbox: MATLAB Toolbox for solving structural, thermal, electromagnetics, and other general PDEs: MathWorks: 3.3 (R2019b) 2019-09-11: Proprietary commercial software
The notation convention chosen here (with W 0 and W −1) follows the canonical reference on the Lambert W function by Corless, Gonnet, Hare, Jeffrey and Knuth. [3]The name "product logarithm" can be understood as follows: since the inverse function of f(w) = e w is termed the logarithm, it makes sense to call the inverse "function" of the product we w the "product logarithm".
Wolfram Alpha: Wolfram Research: 2009 2013: Pro version: $4.99 / month, Pro version for students: $2.99 / month, ioRegular version: free Proprietary: Online computer algebra system with step-by step solutions. Xcas/Giac: Bernard Parisse 2000 2000 1.9.0-99: May 2024: Free GPL: General CAS, also adapted for the HP Prime. Compatible modes for ...
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 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.