Series solutions to differential equations — the alternative method for solving linear differential equations, especially when other methods may not be directly applicable or when finding an exact solution gets you lost in the weeds.
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
By using series solutions, you’ll learn to approximate the solution of a differential equation with a power series containing an infinite number of terms. This approach can be particularly beneficial when tackling higher-order linear differential equations with variable coefficients or when the equation is defined near an ordinary point.
Ordinary Point?! What’s that?! Good question, let’s review some key terms prior to moving forward.
Understanding Ordinary and Singular Points
A point is said to be an ordinary point of a differential equation if, when in standard form, both coefficients in front of y’ and y are analytic. In contrast, a singular point is one that is not an ordinary point.
Defining Analytic Functions
What does it mean to be analytic?
Derivative Formulas for Power Series Solutions
And according to Paul’s Online Notes, functions are analytic if they can be represented by a power series with a positive or infinite radius of convergence. Which means the coefficients are defined for all real numbers.
And a singular point is one that is not an ordinary point.
Significance of Ordinary Points in Series Solutions
Why do ordinary points matter?
Because an ordinary point of a differential equation enables us to find two linearly independent solutions in the form of a power series!
Steps to Solve ODEs Using Series Solutions
How do we infinite series to solve differential equations?
- Substitute our power series into the given equation
- Combine the series by shifting powers and then shifting indices by pulling out initial terms.
- Find a recurrence relation on the coefficients by equating all coefficients to the right-hand side and determine coefficients by solving for higher coefficients in terms of lower ones.
- Use our coefficients to build two solutions by plugging back into our known power series.
Finding a power series solution takes time and organization, so be patient and don’t rush or skip steps.
The cool thing is that once you understand the process, you will be able to solve some complicated second-order ODEs using the method of power series solutions.
Series Solutions – Video
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