On Solving Higher Order Equations for Ordinary ...
Chapter 08.05
On Solving Higher Order Equations
for Ordinary Differential Equations
After reading this chapter, you should be able to:
1. solve higher order and coupled differential equations,
We have learned Euler’s and Runge-Kutta methods to solve first order ordinary differential equations of the form
[pic] (1)
What do we do to solve simultaneous (coupled) differential equations, or differential equations that are higher than first order? For example an [pic]order differential equation of the form
[pic] (2)
with [pic]initial conditions can be solved by assuming
[pic] (3.1)
[pic] (3.2)
[pic] (3.3)
[pic]
[pic] (3.n)
[pic]
= [pic] (3.n+1)
The above Equations from (3.1) to (3.n+1) represent [pic] first order differential equations as follows
[pic] (4.1)
[pic] (4.2)
[pic]
[pic] (4.n)
Each of the [pic]first order ordinary differential equations are accompanied by one initial condition. These first order ordinary differential equations are simultaneous in nature but can be solved by the methods used for solving first order ordinary differential equations that we have already learned.
Example 1
Rewrite the following differential equation as a set of first order differential equations.
[pic]
Solution
The ordinary differential equation would be rewritten as follows. Assume
[pic]
Then
[pic]
Substituting this in the given second order ordinary differential equation gives
[pic]
[pic]
The set of two simultaneous first order ordinary differential equations complete with the initial conditions then is
[pic]
[pic].
Now one can apply any of the numerical methods used for solving first order ordinary differential equations.
Example 2
Given
[pic], find by Euler’s method
a) [pic]
b) the absolute relative true error for part(a), if [pic]
c) [pic]
Use a step size of [pic].
Solution
First, the second order differential equation is written as two simultaneous first-order differential equations as follows. Assume
[pic]
then
[pic]
[pic]
So the two simultaneous first order differential equations are
[pic] (E2.1)
[pic] (E2.2)
Using Euler’s method on Equations (E2.1) and (E2.2), we get
[pic] (E2.3)
[pic] (E2.4)
a) To find the value of [pic] and since we are using a step size of [pic] and starting at [pic], we need to take three steps to find the value of [pic].
For [pic],
From Equation (E2.3)
[pic]
[pic]
[pic]
[pic] is the approximate value of [pic] at
[pic]
[pic]
From Equation (E2.4)
[pic]
[pic]
[pic]
[pic]
[pic]is the approximate value of [pic](same as [pic]) at [pic]
[pic]
For [pic],
From Equation (E2.3)
[pic]
[pic]
[pic]
[pic]
[pic] is the approximate value of [pic] at
[pic]
[pic]
From Equation (E2.4)
[pic]
[pic]
[pic]
[pic]
= 0.31970
[pic]is the approximate value of [pic]at
[pic]
[pic]0
For [pic],
From Equation (E2.3)
[pic]
[pic]
[pic]
[pic]
[pic] is the approximate value of [pic] at
[pic]
[pic]
From Equation (E2.4)
[pic]
[pic]
[pic]
[pic]
[pic]
[pic] is the approximate value of [pic] at
[pic]
[pic]
[pic]
b) The exact value of [pic] is
[pic]
The absolute relative true error in the result from part (a) is
[pic]
= 9.7062%
c) [pic][pic]
Example 3
Given
[pic],
find by Heun’s method
a) [pic]
b) [pic].
Use a step size of [pic].
Solution
First, the second order differential equation is rewritten as two simultaneous first-order differential equations as follows. Assume
[pic]
then
[pic]
[pic]
So the two simultaneous first order differential equations are
[pic] (E3.1)
[pic] (E3.2)
Using Heun’s method on Equations (1) and (2), we get
[pic] (E3.3)
[pic] (E3.4a)
[pic] (E 3.4b)
[pic] (E3.5)
[pic] (E3.6a)
[pic] (E3.6b)
For [pic]
From Equation (E3.4a)
[pic]
[pic]
[pic]
From Equation (E3.6a)
[pic]
[pic]
[pic]
= -4
From Equation (E3.4b)
[pic]
[pic]
[pic]
= 1
From Equation (E3.6b)
[pic]
[pic]
[pic]
[pic]
[pic]
From Equation (E3.3)
[pic]
[pic]
[pic]
[pic] is the approximate value of [pic] at
[pic]
[pic]
From Equation (E3.5)
[pic]
[pic]
[pic]
[pic] is the approximate value of [pic] at
[pic]
[pic]
For [pic]
From Equation (E3.4a)
[pic]
[pic]
[pic]
From Equation (E3.6a)
[pic]
[pic]
[pic]
[pic]
From Equation (E3.4b)
[pic]
[pic]
[pic]
[pic]
From Equation (E3.6b)
[pic]
[pic]
[pic]
[pic]
[pic]
From Equation (E3.3)
[pic]
[pic]
[pic]
[pic] is the approximate value of [pic] at
[pic]
[pic]
From Equation (E3.5)
[pic]
[pic]
[pic]
[pic] is the approximate value of [pic] at
[pic]
[pic]
For [pic]
From Equation (E3.4a)
[pic]
[pic]
[pic]
From Equation (E3.6a)
[pic]
[pic]
[pic]
[pic]
From Equation (E3.4b)
[pic]
[pic]
[pic]
= 0.035836
From Equation (E3.6b)
[pic]
[pic]
[pic]
[pic]
[pic]
From Equation (E3.3)
[pic]
[pic]
[pic]
[pic] is the approximate value of [pic] at
[pic]
[pic]
b) From Equation (E3.5)
[pic]
[pic]
[pic]
[pic] is the approximate value of [pic] at
[pic]
[pic]
The intermediate and the final results are shown in Table 1.
Table 1 Intermediate results of Heun’s method.
|[pic] |0 |1 |2 |
|[pic] |0 |0.25 |0.50 |
|[pic] |1 |1.3750 |1.5738 |
|[pic] |2 |1.1598 |0.55533 |
|[pic] |2 |1.1598 |0.55533 |
|[pic] |[pic] |[pic] |[pic] |
|[pic] |1 |0.43087 |0.035836 |
|[pic] |[pic] |[pic] |[pic] |
|[pic] |1.3750 |1.5738 |1.6477 |
|[pic] |1.1598 |0.55533 |0.13158 |
|ORDINARY DIFFERENTIAL EQUATIONS | |
|Topic |Higher Order Equations |
|Summary |Textbook notes on higher order differential equations |
|Major |General Engineering |
|Authors |Autar Kaw |
|Last Revised |October 13, 2010 |
|Web Site | |
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