The series method is a powerful technique for finding solutions to differential equations, especially when other methods fail. It involves expressing the solution as an infinite power series and then determining the coefficients of the series. When applying this method around an ordinary point, the differential equation's coefficients are analytic, ensuring the existence of a series solution. By substituting the power series into the differential equation and solving for the coefficients recursively, we can find a series representation of the solution.
This worksheet focuses on applying the series method around ordinary points. These points are crucial because they guarantee the existence and convergence of the series solution within a certain interval. Let's work through some vocabulary and practice problems to enhance our understanding of this method.
Match the following terms with their correct definitions:
| Term | Definition |
|---|---|
| 1. Ordinary Point | A. A point where the coefficients of the differential equation are analytic. |
| 2. Power Series | B. An infinite series of the form $\sum_{n=0}^{\infty} a_n (x-c)^n$. |
| 3. Radius of Convergence | C. The distance from the center of the power series to the nearest singularity. |
| 4. Recurrence Relation | D. An equation that defines a sequence recursively: each term is defined as a function of the preceding terms. |
| 5. Analytic Function | E. A function that can be locally given by a convergent power series. |
(Match the numbers 1-5 to the letters A-E)
Complete the following paragraph with the correct words.
When solving a differential equation using the series method around an __________ point, we assume a solution of the form of a __________ __________. We then substitute this series into the differential equation and solve for the __________. The __________ __________ helps us find the relationship between consecutive coefficients.
Explain in your own words why it is important to check the radius of convergence when using the series method to solve differential equations.
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