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15. Show that the solution of the initial value problem my" + yy' + ky = 0, y(to)=yo, y'(to) = y₁ can be expressed as the sum y = v+w, where u satisfies the initial conditions v(to) = yo, v'(to) = 0, w satisfies the initial conditions w(to) = 0, w'(to) = y₁, and both v and w satisfy the same differential equation as u. This is another instance of superposing solutions of simpler problems to obtain the solution of a more general problem.

15. Show that the solution of the initial value problem my" + yy' + ky = 0, y(to)=yo, y'(to) = y₁ can be expressed as the sum y = v+w, where u satisfies the initial conditions v(to) = yo, v'(to) = 0, w satisfies the initial conditions w(to) = 0, w'(to) = y₁, and both v and w satisfy the same differential equation as u. This is another instance of superposing solutions of simpler problems to obtain the solution of a more general problem.

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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15. Show that the solution of the initial value problem my" + y + ky = 0, y(to) = yo, y'(to) = y₁ can be expressed as the sum y = v+w, where v satisfies the initial conditions v(to) = yo, v'(to) = 0, w satisfies the initial conditions w(to) = 0, w'(to) = y₁, and both v and w satisfy the same differential equation as u. This is another instance of superposing solutions of simpler problems to obtain the solution of a more general problem.
Transcribed Image Text:15. Show that the solution of the initial value problem my" + y + ky = 0, y(to) = yo, y'(to) = y₁ can be expressed as the sum y = v+w, where v satisfies the initial conditions v(to) = yo, v'(to) = 0, w satisfies the initial conditions w(to) = 0, w'(to) = y₁, and both v and w satisfy the same differential equation as u. This is another instance of superposing solutions of simpler problems to obtain the solution of a more general problem.
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