Determine two linearly independent solutions to x2y′′+xy′−(4+x)y =0, x > 0.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Determine two linearly independent solutions to x2y′′+xy′−(4+x)y =0, x > 0.
Expert Solution
Step 1
Differential equations are used to describe a system. It's state at every different value of time for different initial values can be solved using a single differential equation. Power series is a method to solve differential equation where the solution is obtained using a recurrence relation.
Given:
Step 2
Assuming power series solution:
Putting this in given equation
Putting coefficient of lowest power equal to zero:
here k can only be positive so taking k=2 series becomes.
Similarly all other coefficients will be also zero so looking at xn+2 we get recurrence relation :
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