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. $4 y'' + 4 y' + 6 y = 0$

The auxiliary equation for the given differential equation $4 y'' + 4 y' + 6 y = 0$ has complex roots and its general solution is $y (t) = e^{- \frac{1}{2} t} (c_{1} \cos (\frac{\sqrt{5} t}{2}) t + c_{2} \sin (\frac{\sqrt{5} t}{2}))$ .

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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:

$y (t) = c_{1} e^{αt} cosβt + c_{2} e^{αt} sinβt$ .

Step 2: Finding the roots of the auxiliary equation.

The given differential equation is $4 y'' + 4 y' + 6 y = 0 .$

Then the auxiliary equation is $4 r^{2} + 4 r + 6 = 0 .$

The roots of the auxiliary equation are:

role="math" localid="1654064879531" $r = \frac{- 4 \pm \sqrt{4^{2} - 4 \times 4 \times 6}}{2 \times 4} r = \frac{- 4 \pm \sqrt{16 - 96}}{} r = \frac{- 4 \pm 4 i \sqrt{5}}{} r = - \frac{1}{2} \pm \frac{\sqrt{5} i}{2}$

Step 3: Final answer.

Therefore, the general solution is:

$y (t) = e^{- \frac{1}{2} \times t} (c_{1} \cos (\frac{\sqrt{5} t}{2} t) + c_{2} \sin (\frac{\sqrt{5} t}{2})) y (t) = e^{- \frac{1}{2} t} (c_{1} \cos (\frac{\sqrt{5} t}{2} t) + c_{2} \sin (\frac{\sqrt{5} t}{2}))$

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

The auxiliary equation for the given differential equation has complex roots. Find a general solution. $4 y'' + 4 y' + 6 y = 0$

Step by Step Solution

Step 1: Complex conjugate roots.

Step 2: Finding the 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

The auxiliary equation for the given differential equation has complex roots. Find a general solution. 4y''+4y'+6y=0

Step by Step Solution

Step 1: Complex conjugate roots.

Step 2: Finding the roots of the auxiliary equation.

Step 3: Final answer.

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The auxiliary equation for the given differential equation has complex roots. Find a general solution. $4 y'' + 4 y' + 6 y = 0$