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 for the differential equation with x as the independent variable. $6 z''' + 7 z'' - z' - 2 z = 0$

Thus, the general solution to the given differential equation is;

z = C_{1} e^{- x} + C_{2} e^{\frac{x}{2}} + C_{3} e^{\frac{- 2 x}{3}}

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Write the auxiliary equation.

The given differential equation is;

$6 z''' + 7 z'' - z' - 2 z = 0$

The auxiliary equation is . $6 m^{3} + 7 m^{2} - m - 2 = 0$

Simplify the auxiliary equation.

$\begin{matrix} 6 m^{3} + 7 m^{2} - m - 2 = 0 \\ 6 m^{3} + 7 m^{2} - m - 2 = 0 \\ (m + 1) (6 m^{2} + m - 2) = 0 \end{matrix}$

One of the roots is . $m_{1} = - 1$

Find the other roots of the auxiliary equation by solving the quadratic equation.

$\begin{matrix} m = \frac{- 1 \pm \sqrt{1 + 4 (12)}}{12} \\ m = \frac{- 1 \pm \sqrt{49}}{12} \\ m = \frac{- 1 \pm 7}{12} \\ m = \frac{- 1 + 7}{12}, \frac{- 1 - 7}{12} \\ m_{2} = \frac{1}{2}, m_{3} = - \frac{2}{3} \end{matrix}$

Step 2: Write the general solution.

The roots are real and distinct; therefore the general solution to the given differential equation is given as:

$\begin{matrix} z = C_{1} e^{m_{1} x} + C_{2} e^{m_{2} x} + C_{3} e^{m_{3} x} \\ z = C_{1} e^{(- 1) x} + C_{2} e^{(\frac{1}{2}) x} + C_{3} e^{(- \frac{2}{3}) x} \\ z = C_{1} e^{- x} + C_{2} e^{\frac{x}{2}} + C_{3} e^{\frac{- 2 x}{3}} \end{matrix}$

Thus, the general solution to the given differential equation is;

$z = C_{1} e^{- x} + C_{2} e^{\frac{x}{2}} + C_{3} e^{\frac{- 2 x}{3}}$

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

Find a general solution for the differential equation with x as the independent variable. $6 z''' + 7 z'' - z' - 2 z = 0$

Step by Step Solution

Step 1: Write the auxiliary equation.

Step 2: Write the general solution.

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Find a general solution for the differential equation with x as the independent variable.6z'''+7z''-z'-2z=0

Step by Step Solution

Step 1: Write the auxiliary equation.

Step 2: Write the general solution.

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Find a general solution for the differential equation with x as the independent variable. $6 z''' + 7 z'' - z' - 2 z = 0$