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Show all the necessary solution steps for each question. 1. (a) Consider the differential equation. 4y" - 4y' + y = 0 Verify that the functions ež and xež form a fundamental set of solutions of the differential equation on the interval (-00,00) (b) Suppose for k ≤n, y₁, y2,..., yk are k linearly independent solutions on (-∞0,00) of a homogeneous linear nth-order differential equation with constant coefficients. Clearly Yk+1=0 is also a solution (al trivial solution) for this homogeneous DE. IS the set of solutions y₁, y2,....., yk, Yk+1 linearly independent or dependent on (-00,00) ? Discuss

Show all the necessary solution steps for each question. 1. (a) Consider the differential equation. 4y" - 4y' + y = 0 Verify that the functions ež and xež form a fundamental set of solutions of the differential equation on the interval (-00,00) (b) Suppose for k ≤n, y₁, y2,..., yk are k linearly independent solutions on (-∞0,00) of a homogeneous linear nth-order differential equation with constant coefficients. Clearly Yk+1=0 is also a solution (al trivial solution) for this homogeneous DE. IS the set of solutions y₁, y2,....., yk, Yk+1 linearly independent or dependent on (-00,00) ? Discuss

Advanced Engineering Mathematics
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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Show all the necessary solution steps for each question. 1. (a) Consider the differential equation. 4y" - 4y' + y = 0 Verify that the functions ež and xež form a fundamental set of solutions of the differential equation on the interval (-∞0,00) (b) Suppose for k ≤n, y₁, y2,..., yk are k linearly independent solutions on (-∞, ∞o) of a homogeneous linear nth-order differential equation with constant coefficients. Clearly Yk+1 = 0 is also a solution (a trivial solution) for this homogeneous DE. IS the set of solutions y₁, y₂,....., yk, Yk+1 linearly independent or dependent on (-00,00) ? Discuss 2. Use reduction of order formula to find the general solution. 2xy" - (2x + 1)y' + y = 0, y₁ = ex
Transcribed Image Text:Show all the necessary solution steps for each question. 1. (a) Consider the differential equation. 4y" - 4y' + y = 0 Verify that the functions ež and xež form a fundamental set of solutions of the differential equation on the interval (-∞0,00) (b) Suppose for k ≤n, y₁, y2,..., yk are k linearly independent solutions on (-∞, ∞o) of a homogeneous linear nth-order differential equation with constant coefficients. Clearly Yk+1 = 0 is also a solution (a trivial solution) for this homogeneous DE. IS the set of solutions y₁, y₂,....., yk, Yk+1 linearly independent or dependent on (-00,00) ? Discuss 2. Use reduction of order formula to find the general solution. 2xy" - (2x + 1)y' + y = 0, y₁ = ex
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