Page 164 - Numerical Analysis Using MATLAB and Excel

Classification

2. Order - The highest order derivative which is included in the differential equation

3. Degree - The exponent of the highest power of the highest order derivative after the differential
equation has been cleared of any fractions or radicals in the dependent variable and its deriva-
tives
For example, the differential equation

3
4
⎛ d y ⎞ 2 ⎛ d y ⎞ 4 ⎛ d y ⎞ 6 ⎛ dy ⎞ 8 y 2 – 2x
2
⎜ --------⎟ 5 --------⎟ ⎜ + 6 --------⎟ ⎜ + + 3 ------ + -------------- = ye
dx ⎠
⎝
3
⎝ dx ⎠ 4 ⎝ dx ⎠ 3 ⎝ dx ⎠ 2 x + 1
is an ordinary differential equation of order and degree . 2
4
y
x
If the dependent variable is a function of only a single variable , that is, if y = f x() , the differ-
ential equation which relates and is said to be an ordinary differential equation and it is abbrevi-
y
x
ated as ODE.
The differential equation
2
d y dy
-------- + 3------ + 2 = 5cos 4t
dt 2 dt
is an ODE with constant coefficients.
The differential equation

2
d y dy 2 2
2
x -------- + x------ + ( x – n ) = 0
dt 2 dt
is an ODE with variable coefficients.

(
,
If the dependent variable is a function of two or more variables such as y = f x t ) , where x
y
and are independent variables, the differential equation that relates , , and is said to be a
t
t
y x
partial differential equation and it is abbreviated as PDE.
An example of a partial differential equation is the well-known one-dimensional wave equation
shown below.
2
2
∂ y 2∂ y
-------- = a --------
∂t 2 ∂x 2
Most engineering problems are solved with ordinary differential equations with constant coeffi-
cients; however, partial differential equations provide often quick solutions to some practical
applications as illustrated with the following three examples.

Numerical Analysis Using MATLAB® and Excel®, Third Edition 5−3

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