Math Problem Statement
Find a Frobenius type solution for the differential equation xy" + y' - y = 0
Solution
To find a Frobenius series solution for the differential equation
we will attempt to find a solution in the form of a power series around :
where is the indicial exponent to be determined, and are the series coefficients.
Step 1: Compute derivatives of the series
First, we calculate the first and second derivatives of the proposed solution.
- The first derivative is:
- The second derivative is:
Step 2: Substitute into the differential equation
We now substitute , , and into the original differential equation:
Substitute each term:
- ( x y'' = x \sum_{n=0}^{\infty} a_n (n+r)(n+r-1) x^{n+r-2} = \sum_{n=0}^{\infty} a_n (n+r)(n+r-1) x^{n+r-
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Math Problem Analysis
Mathematical Concepts
Differential Equations
Frobenius Series
Power Series Solutions
Formulas
y(x) = ∑ a_n x^{n+r}
y'(x) = ∑ a_n (n+r) x^{n+r-1}
y''(x) = ∑ a_n (n+r)(n+r-1) x^{n+r-2}
Theorems
Frobenius Method
Power Series Expansion
Suitable Grade Level
College-Level Mathematics
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