Project
Differential Equations by Power Series
Solve the problems given below, then answer the questions. Calculators and computer algebra systems may be used
for the computations and producing the graphs. The grading criteria are listed on the class web sites.
Analysis of the Issue
For this project you can think of a differential equation as an equation that relates a function to some of its derivatives
and other functions. For some differential equations power series can be used to find the solution. Use the following
as the Taylor series expansion of the function y.
y(x) = a0 + a1 x + a2
x2
x3
x4
+ a3
+ a4
+ ...
2!
3!
4!
Solving y = y 0 when y(0) = 1.
1. Use y(0) = 1 to find a0 .
2. Calculate y 0 (x).
3. Use y = y 0 and your Taylor series to setup an equation.
4. Remembering that two polynomials are equal if and only if all of their coefficients are equal, solve the equation
from the previous step for each of the coefficients ai .
5. Write out the resulting power series (with the ai you just calculated).
6. Identify this function.
Solving y = −y 00 when y(0) = 0 and y 0 (0) = 1.
7. Use y(0) = 0 to find a0 .
8. Calculate y 0 (x).
9. Use y 0 (0) = 1 to find a1 .
10. Calculate y 00 (x).
11. Use y = −y 00 and your Taylor series to setup an equation.
12. Remembering that two polynomials are equal if and only if all of their coefficients are equal, solve the equation
from the previous step for each of the coefficients ai .
13. Write out the resulting power series (with the ai you just calculated).
14. Identify this function.
Solving y ·
x
x+2
= x00 when y(0) = 0, y 0 (0) = 1, and y 00 (0) = 2.
15. Use y(0) = 1 to find a0 .
16. Calculate y 0 (x).
17. Use y 0 (0) = 0 to find a1 .
18. Calculate y 00 (x).
1
Project: Differential Equations by Power Series
2
19. Use y 00 (0) = 2 to find a2 .
20. Use y 0 + y 00 = (2 + 2x + 4x2 )y and your Taylor series to setup an equation.
21. Remembering that two polynomials are equal if and only if all of their coefficients are equal, solve the equation
from the previous step for each of the coefficients ai .
22. Write out the resulting power series (with the ai you just calculated).
23. Can you identify this result? Note in general when using this technique, we do not need to be able to identify
the solution.