Foundations -- Definitions, Order, Degree and Formation

An equation that contains derivatives is called a differential equation (DE).

A normal equation like tells you the direct relationship between and . A differential equation like tells you something about the rate of change of with respect to .

There are two types:

Ordinary Differential Equation (ODE): Only one independent variable is involved. All derivatives are ordinary derivatives ().

Example:

Partial Differential Equation (PDE): More than one independent variable. Derivatives are partial derivatives.

Example:

We will only study ODEs in this course.

Think of it this way: in algebra we solve for a number, but in differential equations we solve for a function. The solution is an entire curve, not a single point.

Order = the highest derivative that appears in the equation.

- If the highest derivative is , the order is 1.

- If the highest derivative is , the order is 2.

- If the highest derivative is , the order is 3.

Degree = the power (exponent) of that highest-order derivative, BUT only after you have removed all roots and fractions from the derivative terms.

Important: If the equation contains or or , the degree is not defined because these are transcendental functions of the derivative.

Always remove radicals and fractions from derivative terms FIRST, then read the degree. The degree of lower-order derivatives does not matter -- we only care about the highest-order derivative.

If a relation between and contains arbitrary constants, we can form a DE by:

1. Differentiating the relation as many times as there are constants

2. Eliminating all the constants

The number of arbitrary constants = the order of the resulting DE.

When we write , it means "square the entire derivative," not . These are completely different things!