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. $y'' - 10 y' + 26 y = 0$

The auxiliary equation for the given differential equation $y'' - 10 y' + 26 y = 0$ has complex roots and its general solution is $y (t) = e^{5 t} (c_{1} \cos (t) + c_{2} \sin (t))$ .

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

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

Step 2: The auxiliary equation.

Assume that $y = e^{rt}$ is a solution to the given equation.

Since the given equation is of order two, differentiate role="math" localid="1654061985491" $y$ with respect to role="math" localid="1654061981089" $x$ twice:

$y' (t) = r e^{rt} y^{"} (t) = r^{2} e^{rt}$

Substitute $y ", y'$ and $y$ in the given equation to obtain: $e^{rt} (r^{2} - 10 r + 26) = 0 .$

Then the auxiliary equation is $r^{2} - 10 r + 26 = 0 .$

Step 3: Finding the roots.

Solve for the roots of the auxiliary equation

$r_{1 / 2} = \frac{10 \pm \sqrt{10^{2} - 4 \times 1 \times 26}}{2 \times 1} = \frac{10 \pm \sqrt{100 - 104}}{} = \frac{10 \pm 2 i}{} = 5 \pm i$

Step 4: Final answer.

Since it has a complex conjugate of the form $r = α \pm iβ$ for $α = 5$ and $β = 1$

Thus, the general solution is $y (t) = e^{5 t} (c_{1} \cos (t) + c_{2} \sin (t))$ .

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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. $y'' - 10 y' + 26 y = 0$

Step by Step Solution

Step 1: Complex conjugate roots.

Step 2: The auxiliary equation.

Step 3: Finding the roots.

Step 4: 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. y''-10y'+26y=0

Step by Step Solution

Step 1: Complex conjugate roots.

Step 2: The auxiliary equation.

Step 3: Finding the roots.

Step 4: Final answer.

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The auxiliary equation for the given differential equation has complex roots. Find a general solution. $y'' - 10 y' + 26 y = 0$