Power Series Solutions
A Differential Equations Approach

Sometimes you’ll encounter cases where it’s not possible to find “nice” or closed-form solutions using the techniques you’ve learned so far.

In such events, power series solutions come to your rescue, offering an alternative approach to finding approximate solutions or solutions that can be expressed as an infinite series very similar to the process we used for Undetermined Coefficients.

Essential Features of Power Series

Now, let’s revisit Power Series from Calculus to understand its importance in solving these complex cases.

Why?

Because a power series has several essential features:

• It is convergent at a specified value of x if its sequence of partial sums converges.
• Every power series has a radius and interval of convergence.
• A power series converges absolutely within its interval of convergence.
• Power series help to define Taylor and Maclaurin Series.
• And, a power series helps us represent functions!

Recalling Power Series from Calculus

Let’s take a step back for a second and remember our calculus days.

Remember how we were able to:

• Represent a function as sums of power series by manipulating geometric series
• Differentiate or integrate a geometric series to find power series representations
• Use the ratio or root tests to determine the radius and interval of convergence
• Understand that Taylor and Maclaurin series are forms of Power series

Eek!

If what I just said scared you or made you scream… “I don’t remember that stuff!”

Then I would greatly recommend that you take a quick look at those Power Series videos before jumping into these two video lessons as they will undoubtedly help give you the foundation you need in finding series solutions for differential equations.

Arithmetic of Power Series

If you didn’t have a total meltdown and remember how to represent a function using Power Series from calculus, then you are ready to tackle the arithmetic of powers series by shifting the summation index.

So, multiple power series can be combined through various operations such as:

• Addition
• Subtraction
• Multiplication
• Division

But when we perform these operations, the powers of x and the indices of our summation will shuffle.

Reindexing Power Series

And it’s our job to reindex them.

How do we do this?

1. Combine our power series by performing the indicated operation.
2. Match the powers of x so they are the same by shifting indices.
3. Shuffle the index to the same starting value by pulling out initial terms.

Confused?

Don’t worry! It will all make sense once you see it in action.

Next Steps

Together, you will learn how to:

• Express a combination of power series as a single power series.
• Find the power series solutions of a linear first-order differential equation whose solutions cannot be written in terms of familiar functions such as polynomials, exponential, or trigonometric functions.

Let’s jump in!

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