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## Series Solutions: Instructor's Guide

### Main Ideas

• Solving differential equations with power series
• Learning how to re-index sums
• Identifying terms that do not appear in more than one sum and what to do with them

Estimated Time: 20 minutes

### Prerequisite Knowledge

Students should have seen at least one example of how to solve a differential equation using the power series method.

### Activity: Introduction

Write the differential equation on the board and ask students to solve it using the power series method. add the differential equation

### Activity: Student Conversations

Places where students struggle (and suggestions for helping them) include:

• How to re-index a sum
1. Use the trick nn + 2 and plug it in everywhere, especially the sum index where you have to solve for n
2. Use a direct substitution and plug it in everywhere (e.g., n = m + 2)
3. Write out the first few terms before and after the substitution to check
• Identifying the coefficient that corresponds to each power
• Writing out the first few terms in the sum
• This is crucial in this example because the sums do not have the same range, so some terms must be pulled out of one. Once they are pulled out, students do not immediately see that the corresponding coefficients must be independently zero.
• This is one of the most difficult parts. Students do not know what they are allowed to choose to be zero and what the equations tell them is always equal to zero.
• Writing the solution in a final form as a series
• Students often think a list of coefficients is the answer; encourage them to write y(z) = … and plug in everything they found
• Making the answer look “pretty”
• For this example, the infinite sum can be rewritten as an exponential. This is not always the case, and a series solution is a perfectly accurate answer at the end

### Activity: Wrap-up

Because students do not have extensive experience with this solution method, it is valuable to work out the full solution on the board (with substantial guidance from the students), stopping to ask students to comment on all or most of the conversation points above. Emphasize places where other groups struggled, and ask students to identify troublesome points and things that were helpful to them so that they can write down tips that will be helpful for solving similar problems in the future.

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