Differential Equations
Chapter 1
1.1
Introduction and Terminology
This section is supposed to calm the nerves of anyone just starting out. It is also very important to build the proper vocabulary to talk about any specialized subject.
A differential equation is an equation that includes derivatives or "differentials".
An ordinary differential equation is ordinary whereas, a partial differential equation involves partial derivatives.
Differential equations can be classified by order. The order of an equation is the highest derivative involved. If a fourth derivative (non-partial) is involved, then the equation is referred to as a fourth-order ordinary differential equation.
Finally we have the very important distinction between linear and non-linear.
Linear differential equations:
- The dependant variable (y) and all of its derivatives are of the first degree.
The power of each term involving y is 1. - Each coefficient depends on only the independent variable x.
If an equation is no a linear equation it is said to be... non-linear.
Solutions
Any function f defined on some interval I, which when substituted into a differential equation reduces the equation to an identity, is said to be a solution of the equation on the interval.
There are Explicit and Implicit solutions. There is also the trivial solution, named because it's zero (frequently a solution to differential equations). Explicit solutions are those which can be expressed in the typical "y equals" format. Implicit solutions are those that cannot be expressed explicitly and should be intuitive.
Typically differential equations that have a solution have an infinite number of solutions. This is because integration is used to solve them and therefore generates at least one constant (c). So a family of curves is generated based on the solution function and on the constant c.
An n-parameter family of solutions are the solutions to an nth-order differential equation. If a solution is free of arbitrary parameters it is called a particular solution. A particular solution may also be obtained using initial conditions in order to define constants that may arise.
Take some time to familiarize yourself with these terms, classifying differential equations, and checking solutions to equations.
