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Q4.3-2E

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Fundamentals Of Differential Equations And Boundary Value Problems
Found in: Page 172
Fundamentals Of Differential Equations And Boundary Value Problems

Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

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Short Answer

The auxiliary equation for the given differential equation has complex roots. Find a general solution y''+y=0.

The auxiliary equation for the given differential equation y''+y=0 has complex roots and its general solution is y=c1cost+c2sint.

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Complex conjugate roots.

If the auxiliary equation has complex conjugate roots α±iβ, then the general solution is given as:

y(t)=c1eαtcosβt+c2eαtsinβt.

Step 2: Finding the roots of the auxiliary equation.

The auxiliary equation for y''+y=0 is k2+1=0 . The solutions of the auxiliary equation are:

k2+1=0 k2=-1 k=±i

Therefore, α=0,β=1

Step 3: Final answer.

Therefore, the general solution is:

y=c1cost+c2sint.

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