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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 6: Q4E (page 337)

use the method of undetermined coefficients to determine the form of a particular solution for the given equation.
$y''' + y'' - 2 y = x e^{x} + 1$

Short Answer

Expert verified

$y_{p} (x) = - \frac{4}{25} x e^{x} + \frac{1}{10} x^{2} e^{x} - \frac{1}{2}$

Step by step solution

01

Find the corresponding auxiliaryequation

Theauxiliary equationof corresponding homogeneous equation

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

The solutions of the auxiliary equation are

$r = - 1 + i, r = - 1 - i, r = 1$

02

Find particular solution

Let the particular solution be

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

Then

$y_{p}^{'} (x) = a e^{x} + (a + 2 b) x e^{x} + b x^{2} e^{x} y_{p}^{''} (x) = (2 a + 2 b) e^{x} + (a + 4 b) x e^{x} + b x^{2} e^{x} y_{p}^{'''} (x) = (3 a + 6 b) e^{x} + (a + 6 b) x e^{x} + b x^{2} e^{x}$

Then

$y_{p}^{'''} (x) + y_{p}^{''} (x) - 2 y_{p} (x) = (3 a + 6 b) e^{x} + (a + 6 b) x e^{x} + b x^{2} e^{x} + (2 a + 2 b) e^{x} + (a + 4 b) x e^{x} + b x^{2} e^{x} - 2 a x e^{x} - 2 b x^{2} e^{x} - 2 c = (5 a + 8 b) e^{x} + 10 b x e^{x} - 2 c$

If $(5 a + 8 b) e^{x} + 10 b x e^{x} - 2 c = x e^{x} + 1$

Then $5 a + 8 b = 0, 10 b = 1 a n d - 2 c = 1$

Then $a = - \frac{4}{25}, b = \frac{1}{10} a n d c = - \frac{1}{2}$

Hence $y_{p} (x) = - \frac{4}{25} x e^{x} + \frac{1}{10} x^{2} e^{x} - \frac{1}{2}$

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Most popular questions from this chapter

use the annihilator method to determinethe form of a particular solution for the given equation.

$y''' + 2 y'' - y' - 2 y = e^{x} - 1$

Use the annihilator method to show that if $a_{0} 鈮� 0$ in equation (4) and $f (x)$ has the form (17) $f (x) = b_{m} x^{m} + b_{m - 1} x^{m - 1} + 鈰� + b_{1} x + b_{0}$ , then $y_{p} (x) = B_{m r} x^{m} + B_{m - 1} x^{m - 1} + 鈰� + B_{1} x + B_{0}$ is the form of a particular solution to equation (4).

Find a general solution to y鈥欌€欌€� - 3y鈥� - y = 0 by using Newton鈥檚 method or some other numerical procedure to approximate the roots of the auxiliary equation.

. find a differential operator that annihilates the given function.

$x e^{- 2 x} + x e^{- 5 x} s i n 3 x$

Find a general solution to

$y^{(4)} + 2 y''' 鈭� 3 y'' 鈭� y' + \frac{1}{2} y = 0$

by using Newton鈥檚 method (Appendix B) or some othernumerical procedure to approximate the roots of the auxiliaryequation.

See all solutions

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