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and te are fundamental solutions of the fol- Problem 1. We know that e lowing equation: d²y dy (1) dx² +2a+ a²y = 0. dx Suppose that we only know one solution e-at of (1). Assume (e-at, y₁ (t)) is a set of fundamental solutions of (1). By Abel's theorem, we know the Wronskian of (1) is given by W (e-at, y₁) = cexp{-f2adt}, use the Wronskian to obtain a first order differential equation of y, and solve it to find the fundamental set of solutions of (1).

and te are fundamental solutions of the fol- Problem 1. We know that e lowing equation: d²y dy (1) dx² +2a+ a²y = 0. dx Suppose that we only know one solution e-at of (1). Assume (e-at, y₁ (t)) is a set of fundamental solutions of (1). By Abel's theorem, we know the Wronskian of (1) is given by W (e-at, y₁) = cexp{-f2adt}, use the Wronskian to obtain a first order differential equation of y, and solve it to find the fundamental set of solutions of (1).

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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Problem 1. We know that eat and te-at are fundamental solutions of the fol- lowing equation: dy (1) d²y dx² +2a- + a²y = 0. da Suppose that we only know one solution e-at of (1). Assume (e-at, y₁ (t)) is a set of fundamental solutions of (1). By Abel's theorem, we know the Wronskian of (1) is given by W (e-at, y₁) = cexp{-f2adt), use the Wronskian to obtain a first order differential equation of y₁ and solve it to find the fundamental set of solutions of (1).
Transcribed Image Text:Problem 1. We know that eat and te-at are fundamental solutions of the fol- lowing equation: dy (1) d²y dx² +2a- + a²y = 0. da Suppose that we only know one solution e-at of (1). Assume (e-at, y₁ (t)) is a set of fundamental solutions of (1). By Abel's theorem, we know the Wronskian of (1) is given by W (e-at, y₁) = cexp{-f2adt), use the Wronskian to obtain a first order differential equation of y₁ and solve it to find the fundamental set of solutions of (1).
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