Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question
**Using Power Series to Solve the Initial-Value Problem** We are tasked with solving the following initial-value problem using a power series approach: \[ y'' + 3xy' + 3y = 0, \] \[ y(0) = 1, \] \[ y'(0) = 0. \] **Solution:** We express the solution \( y \) as a power series: \[ y = \sum_{n=0}^{\infty} a_n x^n + x^2 \sum_{n=0}^{\infty} a_n x^{2n+1}. \] This form allows for solving the differential equation and applying the given initial conditions.
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Transcribed Image Text:**Using Power Series to Solve the Initial-Value Problem** We are tasked with solving the following initial-value problem using a power series approach: \[ y'' + 3xy' + 3y = 0, \] \[ y(0) = 1, \] \[ y'(0) = 0. \] **Solution:** We express the solution \( y \) as a power series: \[ y = \sum_{n=0}^{\infty} a_n x^n + x^2 \sum_{n=0}^{\infty} a_n x^{2n+1}. \] This form allows for solving the differential equation and applying the given initial conditions.
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