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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: Q9E (page 337)

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

Short Answer

Expert verified

$y (x) = \frac{1}{6} x^{3} e^{x} + c_{1} e^{x} + c_{2} x e^{x} + c_{3} x^{2} e^{x}$

Step by step solution

01

Find the corresponding auxiliaryequation

Theauxiliary equationof corresponding homogeneous equation

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

The solutions of the auxiliary equation are

$r = 1, r = 1, r = 1$

Therefore a general solution to the homogeneous equation is

$y_{h} (x) = c_{1} e^{x} + c_{2} x e^{x} + c_{3} x^{2} e^{x}$

02

Find particular solution

Let the particular solution be

$y_{p} (x) = a x^{3} e^{x}$

Then

$y_{p}^{'} (x) = 3 a x^{2} e^{x} + a x^{3} e^{x} y_{p}^{''} (x) = 6 a x e^{x} + 6 a x^{2} e^{x} + a x^{3} e^{x} y_{p}^{'''} (x) = 6 a e^{x} + 18 a x e^{x} + 9 a x^{2} e^{x} + a x^{3} e^{x}$

Then

$y_{p}^{'''} (x) - 3 y_{p}^{''} (x) + 3 y_{p}^{''} (x) - y_{p}^{'} (x) = 6 a e^{x} + 18 a x e^{x} + 9 a x^{2} e^{x} + a x^{3} e^{x} - 18 a x e^{x} - 18 a x^{2} e^{x} - 3 a x^{3} e^{x} + 9 a x^{2} e^{x} + 3 a x^{3} e^{x} - a x^{3} e^{x} \ hfill = 6 a e^{x}$

If $6 a e^{x} = e^{x}$

Then $a = \frac{1}{6}$

Hence $y_{p} (x) = \frac{1}{6} x^{3} e^{x}$

03

Step 3: y(x)=yh+yp

Then $y (x) = \frac{1}{6} x^{3} e^{x} + c_{1} e^{x} + c_{2} x e^{x} + c_{3} x^{2} e^{x}$

Is the general solution of $y''' - 3 y'' + 3 y' - y = e^{x}$

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Most popular questions from this chapter

use the method of undetermined coefficients to determine the form of a particular solution for the given equation.

$y''' + y'' - 5 y' + 3 y = e^{- x} + s i n x$

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.

${\cos 2 x, 鈥� \cos^{2} x, 鈥� \sin^{2} x}$ on $(- 鈭�, 鈥� 鈭�)$

Use the annihilator method to show that if $f (x)$ in (4) has the form $f (x) = a c o s 尾 x + b s i n 尾 x,$

then equation (4) has a particular solution of the form

(18) $y_{p} (x) = x^{s} {A c o s 尾 x + B s i n 尾 x}$ ,where sis chosen to be the smallest nonnegative integer such that $x^{3} c o s 尾 x$ and $x^{3} \sin 尾 x$ are not solutions to the corresponding homogeneous equation

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

$y'' - 6 y' + 9 y = s i n 2 x + x$

In Problems 38 and 39, use the elimination method of Sectionto find a general solution to the given system.

$x - d^{2} y / d t^{2} = t + 1$

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