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Q1E

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Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 337

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

use the method of undetermined coefficients to determine the form of a particular solution for the given equation.
$y''' - 2 y'' - 5 y' + 6 y = e^{x} + x^{2}$

$y_{p} (x) = - \frac{1}{6} x e^{x} + \frac{37}{108} + \frac{5 x}{18} + \frac{x^{2}}{6}$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Find the corresponding auxiliaryequation

Theauxiliary equationof corresponding homogeneous equation

$r^{3} - 2 r^{2} - 5 r + 6 = (r - 1) (r - 3) (r + 2) = 0$

The solutions of the auxiliary equation are

$r = - 2, r = 1 a n d r = 3$

Step 2: Find particular solution

Let the particular solution be

$y_{p} (x) = a x e^{x} + b + c x + d x^{2}$

Then

$y_{p}^{'} (x) = a e^{x} + a x e^{x} + c + 2 d x y_{p}^{''} (x) = 2 a e^{x} + a x e^{x} + 2 d y_{p}^{'''} (x) = 3 a e^{x} + a x e^{x}$

Then

$y_{p}^{'''} (x) - 2 y_{p}^{''} (x) - 5 y_{p}^{'} (x) + 6 y_{p} (x) = 3 a e^{x} + a x e^{x} - 4 a e^{x} - 2 a x e^{x} - 4 d - 5 a e^{x} - 5 a x e^{x} - 5 c - 10 d x + 6 a x e^{x} + 6 b + 6 c x + 6 d x^{2} = - 6 a e^{x} + (6 b - 5 c - 4 d) + (6 c - 10 d) x + 6 d x^{2}$

If $- 6 a e^{x} + (6 b - 5 c - 4 d) + (6 c - 10 d) x + 6 d x^{2} = e^{x} + x^{2}$

Then $- 6 a = 1, 6 b - 5 c - 4 d = 0, 6 c - 10 d = 0, a n d 6 d = 1$

Then $a = - \frac{1}{6}, b = \frac{37}{108}, c = \frac{5}{18} a n d d = \frac{1}{6}$

Hence

$y_{p} (x) = - \frac{1}{6} x e^{x} + \frac{37}{108} + \frac{5 x}{18} + \frac{x^{2}}{6}$

Most popular questions for Math Textbooks

find a general solution to the given equation. $y''' + y'' - 5 y' + 3 y = e^{- x} + s i n x .$

Find a differential operator that annihilates the given function.

(a) x² - 2x + 5

(b) e^3x + x - 1

(c) x sin2x

(d) x²e^-2x cos3x

(e) x² - 2x + xe^-x + sin2x - cos3x

Find a general solution for the differential equation with x as the independent variable.

$y''' + 2 y'' - 8 y' = 0$

Determine the largest interval (a, b) for which Theorem 1 guarantees the existence of a unique solution on (a, b) to the given initial value problem.

$xy''' - 3 y' + e^{x} y = x^{2} - 1 y (- 2) = 1, y' (- 2) = 0, y'' (- 2) = 2$

Find a general solution for the differential equation with x as the independent variable: $y''' - 3 y'' - y' + 3 y = 0$

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