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Q4.3-9E

Expert-verified

Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 172

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. $y'' - 8 y' + 7 y = 0$

The general solution of the given equation $y'' - 8 y' + 7 y = 0$ is $y (t) = c_{1} e^{t} + c_{2} e^{7 t}$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Differentiate the value of y.

Given differential equation is $y'' - 8 y' + 7 y = 0$

Let role="math" localid="1654066682579" $y = e^{rt}$

Therefore,

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

Step 2: Finding the roots of the auxiliary equation.

Then the auxiliary equation is $r^{2} - 8 r + 7 = 0 .$

Now find the roots of the auxiliary equation.

role="math" localid="1654066897417" $r^{2} - 8 r + 7 = 0 r^{2} - r - 7 r + 7 = 0 r (r - 1) - 7 (r - 7) = 0 (r - 1) (r - 7) = 0$

$r = 1, r = 7$

Step 3: Final answer.

Therefore, the general solution is $y (t) = c_{1} e^{t} + c_{2} e^{7 t}$

Most popular questions for Math Textbooks

Decide whether or not the method of undetermined coefficients can be applied to find a particular solution of the given equation. $2 y'' (x) - 6 y' (x) + y (x) = \frac{sinx}{e^{4 x}}$

Find the solution to the initial value problem.

$y'' + y' - 12 y = e^{t} + e^{2 t} - 1; y (0) = 1, y' (0) = 3$

A mass–spring system is driven by a sinusoidal external force $g (t) = 5 sint$ . The mass equals 1, the spring constant equals 3, and the damping coefficient equals 4. If the mass is initially located at $y (0) = \frac{1}{2}$ and at rest, i.e., $y' (0) = 0$ , find its equation of motion.

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$y'' (θ) - 7 y' (θ) = θ^{2}$

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