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

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.)
$y'' + 3 y' - 7 y = t^{4} e^{t}$

The particular solution is $y_{p} (x) = (A_{4} t^{4} + A_{3} t^{3} + A_{2} t^{2} + A_{1} t + A_{0}) e^{t}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Use the method of undetermined coefficients to find a particular solution of a given differential equation.

The given differential equation is in the form of;

$ax'' + bx' + cx = e^{rt}$

According to the method of undetermined coefficients, to find a particular solution to the differential equation:

$ay'' (x) + by' (x) + cy (x) = {Ct}^{m} e^{rt}$

Where m is a non-negative integer, use the form

$y_{p} (x) = t^{s} (A_{m} t^{m} + . .. + A_{1} t + A_{0}) e^{rt}$

s = 0 if r is not a root of the associated auxiliary equation;
s = 1 if r is a simple root of the associated auxiliary equation;
s = 2 if r is a double root of the associated auxiliary equation.

Step 2: Now, write the auxiliary equation of the above differential equation

The given differential equation is,

$y'' + 3 y' - 7 y = t^{4} e^{t} . ..... (1)$

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

$y'' + 3 y' - 7 y = 0$

The auxiliary equation for the above equation,

$r^{2} + 3 r - 7 = 0$

Step 3: Now find the roots of the auxiliary equation

Solve the auxiliary equation,

$\begin{matrix} r^{2} + 3 r - 7 = 0 \\ r = \frac{- 3 \pm \sqrt{3^{2} - 4 (1) (- 7)}}{2 (3)} \\ r = \frac{- 3 \pm \sqrt{37}}{6} \end{matrix}$

The roots of the auxiliary equation are;

$r_{1} = \frac{- 3 + \sqrt{37}}{6}, r_{2} = \frac{- 3 - \sqrt{37}}{6}$

The complementary solution of the given equation is;

$y_{c} = c_{1} e^{\frac{- 3 + \sqrt{37}}{6} t} + c_{2} e^{\frac{- 3 - \sqrt{37}}{6} t}$

Step 4: Final conclusion.

To find a particular solution to the differential equation

$ay'' (x) + by' (x) + cy (x) = {Ct}^{m} e^{rt}$

Compare with the given differential equation,

$y'' + 3 y' - 7 y = t^{4} e^{t}$

Condition satisfied,

M=4, s = 0 if is not a root of the associated auxiliary equation;

Therefore, the particular solution of the equation,

$\begin{matrix} y_{p} (x) = t^{0} (A_{4} t^{4} + A_{3} t^{3} + A_{2} t^{2} + A_{1} t + A_{0}) e^{t} \\ y_{p} (x) = (A_{4} t^{4} + A_{3} t^{3} + A_{2} t^{2} + A_{1} t + A_{0}) e^{t} \end{matrix}$

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.)
$y'' + 3 y' - 7 y = t^{4} e^{t}$

Step by Step Solution

Step 1: Use the method of undetermined coefficients to find a particular solution of a given differential equation.

Step 2: Now, write the auxiliary equation of the above differential equation

Step 3: Now find the roots of the auxiliary equation

Step 4: Final conclusion.

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Fundamentals Of Differential Equations And Boundary Value Problems

Answers without the blur.

Short Answer

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.)y''+3y'-7y=t4et

Step by Step Solution

Step 1: Use the method of undetermined coefficients to find a particular solution of a given differential equation.

Step 2: Now, write the auxiliary equation of the above differential equation

Step 3: Now find the roots of the auxiliary equation

Step 4: Final conclusion.

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Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.)
$y'' + 3 y' - 7 y = t^{4} e^{t}$