2.1.11.8 Book 2 Background .1 Cosmology .11 Math Section 8.2

#### Book 9 Computational Mathematics Chapter 8 Ordinary Differential Equation Initial Value Problems

Section 1 General Remarks. Cauchy's Problem.

Cauchy's Problem, the Initial Value Problem (I.V.P.) for ordinary differential equations of order n

(1.1)

involves the determination of the function y = y(x) which satisfies this equation and initial conditions

(1.2)

 where x0, y ' , ???, y (n-1) have given values. 0 0

The Cauchy Problem for the system of differential equations

(1.3)

involves the determination of those functions which satisfy this system of equations and the initial conditions:

(1.4)

Any system which contains higher order derivatives can be transformed into the system (1.3). In particular, the differential equation of order n

is reduced to the form of (1.3) with the aid of the substitutions

,

a process which yields the system:

If the general solution of Equation (1.1) or System (1.3) has been found, then the Cauchy Problem is reduced to finding values of arbitrary constants. But it is rather difficult to find exact solutions of Cauchy Problems; more often than not approximate methods of solution are employed. According to the form in which solutions are sought, approximate methods are subdivided into the two groups:

1. Analytic methods which yield an approximate, analytic solutions.

2. Numerical methods which yield values of an approximate solution in the form of a table. Throughout the sequel, it is assumed that solutions of the equations under consideration exist and are unique.