# Finding roots of equations involving functions defined by first order algebraic differential equations

• Daniel Richardson
Chapter
Part of the Progress in Mathematics book series (PM, volume 94)

## Abstract

A method is given for approximating solutions of f (x) = 0, for f in a certain class F of real valued analytic functions of one real variable. The method depends on being able to decide the sign of f (r) for given f in F and rational r. That is, approximation of roots is Turing reduced to the constant problem. The last root problem is evaded: essentially solutions are only found in given finite intervals.

The class of functions F contains x, exp(x), log(x) for x > 0, sin(x) for x in (—π/2, π/2), and is closed under field operations, differentiation and integration. F is built up by successively adding solutions g of first order differential equations q(x, g, g′) = 0, where q is a polynomial whose coefficients may involve previously defined functions.

The main technique used is construction of a false derivative. A false derivative of f (x) is a function f *(x) which is continuous and has the same sign as f′(x) whenever f(x) = 0.

## Keywords

Normal Form Order Differential Equation Finite Interval Algebraic Differential Equation Quotient Field
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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