Question

(a) Seek power series solutions of the given differential equation about the given point x₀; find the recurrence relation.

(b) Find the first four terms in each of two solutions y₁ and y₂ (unless the series terminates sooner).
(c) By evaluating the Wronskian W( y₁ , y₂) (x₀) show that y₁ and y₂ form a fundamental set of solutions.
(d) If possible, find the general term in each solution.

Answer 1

This mathematics problem involves finding power series solutions to a differential equation, deriving a recurrence relation, calculating the first four terms of the solutions, proving the solutions form a fundamental set by evaluating the Wronskian, and possibly finding the general term of the series solutions.

Step-by-step explanation:

To answer this question, we will be addressing a mathematics problem related to power series solutions of differential equations. The problem asks for the following:

1. Finding power series solutions and a recurrence relation for a differential equation around a given point x₀.
2. Calculating the first four terms of the two solutions y₁ and y₂, unless the series terminates earlier.
3. Using the Wronskian W(y₁, y₂)(x₀) to prove that y₁ and y₂ are a fundamental set of solutions.
4. If possible, determining the general term for each solution.

For part (a), the approach would involve substituting a power series into the differential equation and then equating coefficients for each power of x to derive a recurrence relation. In part (b), we would explicitly calculate the first few terms using the recurrence relation.

Confirming that y₁ and y₂ form a fundamental set involves calculating the Wronskian and showing it is non-zero at the given point x₀ as mentioned in part (c). For part (d), finding a general term would typically require discerning a pattern from the recurrence relation and calculated terms.