Theorem 3.2.7 (Abel's Theorem)4If y1 and y2 are solutions of the second-order linear differential equation(22)L[y]=y′′+p(t)y′+q(t)y=0where p and q are continuous on an open interval I, then the Wronskian W[y1, y2](t) is given by(23)W[y1,y2](t)=cexp(−∫p(t)dt)where c is a certain constant that depends on y1 and y2, but not on t. Further, W[y1, y2](t) either is zero for all t in I (if c = 0) or else is never zero in I (if c ≠ 0).

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Answered: Theorem 3.2.7 (Abel's Theorem)4 If y1…

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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Theorem 3.2.7 (Abel's Theorem)⁴

If y₁ and y₂ are solutions of the second-order linear differential equation

(22)

L[y]=y′′+p(t)y′+q(t)y=0

where p and q are continuous on an open interval I, then the Wronskian W[y₁, y₂](t) is given by

(23)

W[y1,y2](t)=cexp(−∫p(t)dt)

where c is a certain constant that depends on y₁ and y₂, but not on t. Further, W[y₁, y₂](t) either is zero for all t in I (if c = 0) or else is never zero in I (if c ≠ 0).

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