Introduction to Differential Equations - The Engineer's Reference

[Pages:8]Introduction to Differential Equations

HOW TO CLASSIFY A DIFFERENTIAL EQUATION

What is a differential equation

Differential Equation:

And equation containing the derivatives of one or more unknown functions (or dependent variables), with respect to one or more independent variables.

Classified by:

Type Order Degree Linearity

Differential Equations

No derivative

Normal form:

-

+ - - + + + = ()

nth derivative

1st derivative

Forced function

Unknown function and its derivatives

Types of Differential Equations (DE)

Ordinary differential equation (ODE): One (1) independent variable + =

Partial differential equation (PDE): Two (2) or more variables along with partial derivatives + - =

DE Order

Order of DE: Highest derivative that appears in the equation

+ = Second order

Derivative

2 2

Order 1st 2nd 3rd 1st

2nd

Degree of a differential equation

Degree: The exponent of the highest derivative occurring in it after the equation has been rationalized with respect to the highest derivative

=

+

=

+

- =

? Degree of 1 ? Degree of 1 - Degree of 2

Linearity

Linearity: In DE an equation is linear if there is no compounding influence on the dependent variable or any of this derivatives.

Identifying a linear vs non-linear DE

1. The dependent variable and its derivatives are raised to the power 1 2. The coefficients depend, at most, on the independent variable

Normal form:

+

-

- -

+

+

+

= ()

Example: + 32 + ln = 0

Non-linear

Summary

Type:

Order:

Degree: Linearity:

(ODE) One (1) independent variable (PDE) Two (2) or more variables

Highest derivative that appears in the equation

The exponent of the highest derivative

+

+ =

+ + = 2nd Order

+ + = 3rd degree

There is no compounding influence on the dependent variable or any of this

derivatives

+ + = Nonlinear

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