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

Find a general solution. $y'' + 2 y' + 5 y = 0$

The general solution of the given equation $y'' + 2 y' + 5 y = 0$ is $y (t) = e^{- t} (c_{1} \cos (2 t) + c_{2} \sin (2 t))$ .

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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 $α \pm i β$ , then the general solution is given as:

$y (t) = c_{1} e^{α t} c o s β t + c_{2} e^{α t} s i n β t$ .

Step 2: Finding roots of the auxiliary equation.

Given differential equation is $y'' + 2 y' + 5 y = 0 .$

Then the auxiliary equation is $r^{2} + 2 r + 5 = 0 .$

The roots of the auxiliary equation are:

role="math" localid="1654070409803" $r = \frac{- 2 \pm \sqrt{2^{2} - 4 \times 1 \times 5}}{2 \times 1} r = \frac{- 2 \pm \sqrt{4 - 20}}{2} r = \frac{- 2 \pm \sqrt{- 16}}{2} r = \frac{- 2 \pm 4 i}{2} r = - 1 \pm 2 i$

Step 3: Final answer.

Therefore, the general solution is:

$\begin{matrix} y (t) = e^{- 1 \times t} (c_{1} \cos (2 t) + c_{2} \sin (2 t)) \\ = e^{- t} (c_{1} \cos (2 t) + c_{2} \sin (2 t)) \end{matrix}$

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Find a general solution. $y'' + 2 y' + 5 y = 0$

Step by Step Solution

Step 1: Complex conjugate roots.

Step 2: Finding roots of the auxiliary equation.

Step 3: Final answer.

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Find a general solution. y''+2y'+5y=0

Step by Step Solution

Step 1: Complex conjugate roots.

Step 2: Finding roots of the auxiliary equation.

Step 3: Final answer.

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Find a general solution. $y'' + 2 y' + 5 y = 0$