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Examples--Series
Solutions
It is recommended that you come to class since
I will explain many of these examples in more detail in class.
Maple Worksheet on
graphing functions defined by power series using finite series approximations
In the examples below yH and yc
stand for the same thing (the complimentary
solution which is the solution to the corresponding Homogeneous
equation). I apologize for this inconsistency. The examples were
constructed at different times while using different textbooks. I may some
day get around to fixing the inconsistency.
| A Maclaurin Polynomial
Approximation To A Solution
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Maclaurin Polynomial Approximation in Blue
Maple Worksheet
Analytical Solution in Red:
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Series Solution Initial Conditions
Below are the relationships between the
initial conditions and ao, a1, and a2 if the
series represents the whole solution (and not just yc with yp
found separately not using series) and the series solution is centered at the same value for x as the x
value where the initial conditions are prescribed (0 in the case shown below) .
Series Solution Example 0--A Teaching Example That
Does Not Require The Series Solution Method
| The first order differential equation
given below has been solved previously in a number of ways. Here is
an approach that is much more involved than any of the earlier methods but
this example can be used as an introduction.
The complementary solution (or complementary
function) could be found as follows:
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Series Solution Example 1--A Teaching Example That
Does Not Require The Series Solution Method
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Thus
the solution is y = sin(x). Here is an
audio/video illustrating how the
series solution found in Example 1 above could be evaluated for a
particular value of x using a TI-89 or a TI-84.
Click here to see a visual of the
TI-89 commands shown on the video. The video can be paused to
study the TI-84 commands.
Here is a
short PowerPoint
presentation showing the 1st through the 23rd Maclaurin polynomials
approximating the sine function and a
slightly longer version. Here is an applet that will graph
Taylor approximations to
the sine function and another one that will graph Taylor
approximations to exp(x). (The Taylor polynomials are centered at
zero.) We also have a nice Quicktime
animation that looks at Maclaurin polynomials approximating f(x) =
sin(x) + cos(x). Below are the first,
third,
fifth,
seventh,
ninth,
eleventh, and
thirteenth Maclaurin polynomials
approximating the sine function (sine function in blue).
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Series Solution Example 2
This is a nice one.
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| Series Solution Example 3
An example with initial conditions
not prescribed at 0
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The recurrence formula could be used to generate
as many terms in the solution series as are needed for a prescribed level
of accuracy. The radius of convergence of the power series
solution is at least 5. Below is the sixth degree polynomial
approximation to the solution and its graph. Adding
a few more terms does not have a lot of effect as can be seen in the Maple
worksheet linked to below. The excel table linked to below
illustrates the coefficients an, a(n) in the table, (a8
actually equals 0) and the approximate y-values for various values of v.
For each v-value the values in the column above it show the approximate
y-value for each n (i.e., by summing up to that value of n). We
can observe that the series appears not to converge for v > 5.
The TI-84 video below may be easier to understand after watching the TI
video below Example 1. This one is shorter. 
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Maple
Worksheet
Excel Worksheet (The
Making of the Excel Worksheet)
TI-84 Audio/Video |
| Series Solution Example 4
A non-homogeneous example
To find ao and a1 we need
to find yp and use the initial conditions. Since the DE
does not have constant coefficients we cannot use our methodical method
for Undetermined Coefficients. However, inspection of the
coefficients of y and y' might lead us to try
yp=Asin(x).
This would lead to
-Asin(x) - 4Axcos(x) +Axsin(x) = -sin(x) -
4xcos(x) + xsin(x)
What good fortune. A =
1 and yp = sin(x)
Thus
Notice how nice it would be if y(0) = 0. In
this case ao = 0, a1 = 0, and a2 = 0 so
all an = 0 and the solution satisfying the initial conditions
is
y = yp = sin(x)
Maple Worksheet
EC: Reconcile the solution given on the
Maple worksheet with y(0) =1, y'(0) = 1 to the solution indicated above. |
| Series Solution Example 5
A third order DE
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| Series Solution Example 6
Another nonhomogeneous example
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Series
Solution Example 7
a2
and a3 must be zero
| Series Solution Example 8--One of
Airy's Equations
Here
is a Maple Worksheet with truncated series
solutions and their graphs. |
The
figure below shows the graphs of the truncated solutions to the example of Airy's equation solved
on the left for
 (blue)
 (red)
Click
here to view an applet that will show you more on the graphs of this
version of Airy's Equation. This equation "is encountered in
the study of diffraction of light, diffraction of radio waves around the
surface of the earth, aerodynamics, and the deflection of a uniform thin
vertical column that bends under its own weight" (Zill, page 274).
 (red)
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The
differential equation in example 9 arises in the treatment of the
harmonic oscillator in quantum mechanics.
| Series Solution Example 9
Some Hermite Polynomials
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Example 9 Variation Finding an+2
In Terms Of an
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| On
the left below is a graph of the solution to the EC above. The two
EC problems above are the same.
EC2:
Change the initial conditions and find a value for k in the problem solved
above so that the solution will be a polynomial of degree 4.
On the
right below is the graph of one possible solution (perhaps the most
obvious one) to EC2. |
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Series
Solution Example 10--Nonhomogeneous Example--Three Variations Including Finding an+2
In Terms Of an
Variation
1--This Is Similar to the Approach in Example 6 (Finding yp First). 
Note:
The yc
above is actually what we have left of yc after we substitute
in the values we get for ao and a1 from the initial
conditions.
Variation 2--Look for the Whole Solution all at
Once--No Searching for yp. (This is what your textbook does.)
Variation 3--Use the
Repeated Differentiation Method to Find the Maclaurin Polynomial Solution.
Series
Solution Example 11
Trying
something a little different
| Series
Solution Example 12
You need to be able to show the differentiation
steps to produce this approximate solution that would be pretty accurate
for x close to zero. Click here
for a Maple worksheet showing additional terms in the series solution and
graphs.
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Some Differentiation Steps
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| Series Solution Example 13
Find the first three nonzero terms in a series
solution to the given initial value problem.
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Click on the picture to enlarge. The picture
shows the graphs of MacLaurin polynomial approximate solutions of degree
5 (red), 9, 11, 15, 23, 35, 59, and 85 (black).
Maple Worksheet |
| Extra Credit
Hint |
The exact analytical solution is graphed in red
and the solution represented by the first seven terms in the power series
is graphed in blue. |
Series
Solution to a Second Order Nonlinear Equation
Take
a look at this Maple worksheet investigating a
series solution to a second order, nonlinear, initial value problem where the
exact analytical solution is known and available for comparison.
return
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