TABLE OF CONTENTS Differential equations are equations in which unknowns are functions of one or more variables rather than variables themselves (i.e., numbers). Such equations (or systems) include relationships between unknown functions and their derivatives. If equations include derivatives by only one variable, they are called ordinary differential equations; otherwise we are dealing with partial differential equations ( see Chapter 12). Thus, solving (or integrating) a differential equation involves finding an unknown function within the specified interval of values taken by its variables.
It is a well-known fact that one ordinary differential equation ( see Sections 11.1 and 11.2) or system of ordinary differential equations ( see Section 11.3) has a single solution provided that besides the equation itself, we have appropriately specified the initial or boundary conditions. In higher mathematics, the existence and uniqueness of solution theorems have been proven in accordance with specific conditions. Mathcad 2001i provides the tools to solve the following two types of tasks:
Cauchy problems - These are problems for which initial conditions for unknown functions are specified, i. e., these functions' values are set in the initial point of the equation's integration interval.
Boundary value problems - These are problems for which specific conditions on both boundaries of the interval are specified (these problems will be considered in Chapter 12).
As a rule, solving Cauchy problems for ordinary differential equations and systems of such equations is thoroughly studied, and in terms of calculation, this task doesn't present serious...
