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

In Problems 35, use the method of undetermined coefficients to find a particular solution to the given higher-order equation. $y''' + y'' - 2 y = {te}^{t}$

The particular solution is $y_{p} (t) = t (\frac{1}{10} t - \frac{4}{25}) e^{t} .$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Firstly, write the auxiliary equation of the given differential equation.

The given differential equation is:

$y''' + y'' - 2 y = {te}^{t} \dots (1)$

Write the homogeneous differential equation of the equation (1),

$y''' + y'' - 2 y = 0$

The auxiliary equation for the above equation,

$m^{3} + m^{2} - 2 = 0$

Solve the auxiliary equation,

role="math" localid="1654925783487" $\begin{matrix} (m - 1) (m^{2} + 2 m + 2) = 0 \\ m = 1, m = \frac{- 2 \pm \sqrt{4 - 8}}{2} \\ m = 1, m = - 1 \pm i \end{matrix}$

Step 2: Use the method of undetermined coefficients to find a particular solution to a given differential equation.

Consider the particular solution is,

$y_{p} (t) = t (At + B) e^{t} \dots (2)$

Take the first, second, and third derivative of the above equation,

$\begin{matrix} y_{p}' (t) = ({At}^{2} + (2 A + B) t + B) e^{t} \\ y_{p}'' (t) = ({At}^{2} + (4 A + B) t + (2 A + 2 B)) e^{t} \\ y_{p}''' (t) = ({At}^{2} + (6 A + B) t + (6 A + 3 B)) e^{t} \end{matrix}$

Substitute value of $y_{p} (t), y_{p}'' (t)$ and $y_{p}''' (t)$ in the equation (1),

$\begin{matrix} y''' + y'' - 2 y = {te}^{t} \\ ({At}^{2} + (6 A + B) t + (6 A + 3 B)) e^{t} + ({At}^{2} + (4 A + B) t + (2 A + 2 B)) e^{t} - 2 ({At}^{2} + Bt) e^{t} = {te}^{t} \\ [10 At + (8 A + 5 B)] e^{t} = {te}^{t} \end{matrix}$

Comparing the coefficients of the above equation;

role="math" localid="1654925973361" $\begin{matrix} 10 A = 1 \\ A = \frac{1}{10} \\ 8 A + 5 B = 0 . .. (3) \end{matrix}$

Substitute the value A in the equation (3),

$\begin{matrix} 8 (\frac{1}{10}) + 5 B = 0 \\ \frac{4}{5} + 5 B = 0 \\ B = - \frac{4}{25} \end{matrix}$

Step 3: Conclusion.

Substitute values A and B in the equation (2),

$\begin{matrix} y_{p} (t) = t (At + B) e^{t} \\ y_{p} (t) = t (\frac{1}{10} t - \frac{4}{25}) e^{t} \end{matrix}$

Therefore, the particular solution of the equation (1),

$y_{p} (t) = t (\frac{1}{10} t - \frac{4}{25}) e^{t}$

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

In Problems 35, use the method of undetermined coefficients to find a particular solution to the given higher-order equation. $y''' + y'' - 2 y = {te}^{t}$

Step by Step Solution

Step 1: Firstly, write the auxiliary equation of the given differential equation.

Step 2: Use the method of undetermined coefficients to find a particular solution to a given differential equation.

Step 3: Conclusion.

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

In Problems 35, use the method of undetermined coefficients to find a particular solution to the given higher-order equation.y'''+y''-2y=tet

Step by Step Solution

Step 1: Firstly, write the auxiliary equation of the given differential equation.

Step 2: Use the method of undetermined coefficients to find a particular solution to a given differential equation.

Step 3: Conclusion.

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In Problems 35, use the method of undetermined coefficients to find a particular solution to the given higher-order equation. $y''' + y'' - 2 y = {te}^{t}$