Search results
Results from the WOW.Com Content Network
The last equation involves only one unknown. Solving it in turn reduces the next last equation to one unknown, so that this backward substitution can be used to find all of the unknowns: = ′ = ′ ′ +; =,, …,
The substitution that is needed to solve this Bernoulli equation is = Substituting = + directly into the Riccati equation yields the linear equation ′ + (+) = A set of solutions to the Riccati equation is then given by = + where z is the general solution to the aforementioned linear equation.
A unification problem is a finite set E={ l 1 ≐ r 1, ..., l n ≐ r n} of equations to solve, where l i, r i are in the set of terms or expressions.Depending on which expressions or terms are allowed to occur in an equation set or unification problem, and which expressions are considered equal, several frameworks of unification are distinguished.
are equivalent to Newton's equations for the function =, where T is the kinetic, and V the potential energy. In fact, when the substitution is chosen well (exploiting for example symmetries and constraints of the system) these equations are much easier to solve than Newton's equations in Cartesian coordinates.
Let y (n) (x) be the nth derivative of the unknown function y(x).Then a Cauchy–Euler equation of order n has the form () + () + + =. The substitution = (that is, = (); for <, in which one might replace all instances of by | |, extending the solution's domain to {}) can be used to reduce this equation to a linear differential equation with constant coefficients.
One method of solving elementary functional equations is substitution. [citation needed] Some solutions to functional equations have exploited surjectivity, injectivity, oddness, and evenness. [citation needed] Some functional equations have been solved with the use of ansatzes, mathematical induction. [citation needed]
The identity substitution, which maps every variable to itself, is the neutral element of substitution composition. A substitution σ is called idempotent if σσ = σ, and hence tσσ = tσ for every term t. When x i ≠t i for all i, the substitution { x 1 ↦ t 1, …, x k ↦ t k} is idempotent if and only if none of the variables x i ...
In computer science and mathematical logic, satisfiability modulo theories (SMT) is the problem of determining whether a mathematical formula is satisfiable.It generalizes the Boolean satisfiability problem (SAT) to more complex formulas involving real numbers, integers, and/or various data structures such as lists, arrays, bit vectors, and strings.