# Section 2.1: Introduction to Equations

## Equations and Reality

There are several different kinds or categories of equation, all of which require
a different approach to their solution. Indeed, the fundamental nature of the
solution differs significantly for the various categories.
There is also a connection between the physical system and the equations which
describe it, in that certain types of equation describe certain
situations or phenomena. An understanding of the relationship between
the `real' situation and the types of equations is thus essential in
developing mathematical models in the form of
equations.

## Classification of Equations and Systems

- What kinds of equation are there?
- How do these relate to engineering situations?
- What is the nature of the solution?

### What is an equation?

An equation is anything with an equals sign in it, i.e.:

(Something) = (Something else)
There are ** three ** major categories of equation.

### Algebraic Equations

These are most easily identifiable by what they do
**not** contain, i.e. ** No derivatives or integrals**
but any of the normal arithmetic operators or higher algebraic functions.
The following are thus all algebraic equations:
**
***x * - 3 = 5
sin 3` x ` = `y` cos ` x `

`
a x`^{2} + b x + c = 0

**
**
The solution is a numerical value for a single equation, or a set of
numbers, as many as there are unknowns and equations, for a set of algebraic
equations.

One significant characteristic of an equation is whether it
is ** linear ** or ** nonlinear ** in a particular variable.
The equation is linear only if the variable appears to a power
of one, and does not appear as the argument
of a higher function.

Thus only the first equation above is linear in ` x`.
However the second equation is linear in `y`, and the
third is linear in ` a, b, ` and ` c.`

There is
more about algebraic equations
in section 3.1.

### Ordinary Differential Equations: o.d.es

These are readily identifiable as they
contain derivatives:

**NB: these must all be `Straight d' derivatives if the equations are
o.d.e.s. **

There are two different kinds of variable in this type of equation:
the dependent and independent variables.
There may only be ** one independent variable, **
and this will appear on the bottom line of the derivatives.
The variables which appear on the top line are the
** dependent variables ** and there should be the same number of these as
there are equations.

The solution is the dependent variable or variables
as a function or set of functions in terms of the independent variable.

There is
more about o.d.e.s in section 4.1.

### Partial Differential Equations: p.d.e.s

If there is more than one independent variable, then
derivatives must normally be written as partial
derivatives, giving rise to p.d.e.s with `curly ds', e.g.:

P.d.e.s are hard to solve! The solution is the dependent variables as functions
of all the independent variables.

Next - Section 2.2: Examples

Move back to Section 2 Index