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Solution Manual for Advanced Engineering Mathematics, 8th Edition by O'Neil – Complete Chapter 2 Solutions (Second-Order Differential Equations, Wronskian, Euler Equations, Series & Frobenius Methods)

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Pass your engineering math course with the official Solution Manual for Advanced Engineering Mathematics, 8th Edition by Peter V. O'Neil – this instant digital download provides complete, verified step-by-step solutions for all problems in Chapter 2: Second-Order Differential Equations, including linear second-order equations, Wronskian and linear independence, constant coefficient homogeneous equations, particular solutions using undetermined coefficients, variation of parameters, Euler differential equations, power series solutions, Frobenius method, indicial equations, recurrence relations, initial value problems, complementary and particular solutions, method of undetermined coefficients for polynomials, exponentials, sines and cosines, and real-world applications. Perfect for engineering students, mathematics majors, physics students, and anyone preparing for exams in differential equations, advanced engineering mathematics, or applied math. Covers homogeneous and nonhomogeneous ODEs, distinct real roots, repeated roots, complex roots, Euler-Cauchy equations, series solutions near ordinary points, Frobenius solutions near regular singular points, and boundary value problems. Download now and master second-order differential equations today! 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Solution Manual for Advanced Engineering Mathematics 8th
Edition ONeil

Chapter 2

Second-Order Di erential
Equations

2.1 The Linear Second-Order Equation
g; g; g;

1. It g; is g; a g;routine g;exercise g;in g; di g;erentiation g;to g;show g;that g ; y1(x) g; and g; y2(x) g;are
solutions g;of g;the g;homogeneous g;equation, g;while g;yp(x) g;is g;a g;solution g;of
g;the g;nonhomogeneous g;equation. g;The g;Wronskian g;of g;y1(x) g;and

g;y2(x) g;is
W g;(x) g;= 6 g;cos(6x) g ; g ; 6 g;sin(6 g ; x) g;= g;6 g;sin g ; 2(x) g ; 6 g;sin2(x) g;= g;6;
sin(6x) cos(6x)

and g;this g;is g;nonzero g;for g;all g;x, g;so g;these g ; solutions g;are g;linearly
g;independent g;on g;the g;real g;line. g;The g;general g;solution g;of g;the

g;nonhomogeneous g;di g;erential g;equation g;is

1
y g;= g;c1 g;sin(6x) g;+ g;c2 g;cos(6x) g;+ g;36g(; x
g ; 1): g;For g;the g;initial g;value g;problem, g;we g;need

1
y(0) g;= 36 g ; = g;5
g;c2
so g;c2 g;= g;179=36.
g;And

1
y0(0) g;= g;2 g;= g;6c1 g;+ 36
g;
g;

so g;c1 g;= g;71=216. g;The g;unique g;solution g;of g;the g;initial g;value g;problem g;is
71 g; 179 1
y(x) = 216 sin(6x) 36 g;cos(6x) g;+ g;36 g ; g ; (x g;1):
2. The g;W ronskian g;of g;e4x g;and g;e4 g ; x g;
is
W g;(x) g;= 4e4x 4e 4 g; x =8 g ; 6=0
e4x e4 x

37

© g;2018 g;Cengage g;Learning. g;All g;Rights g;reserved. g;May g;not g;be g;scanned, g;copied g;or g;duplicated, g;or g;posted g;to g;a g;publicly g;accessible g;website, g;in g;whole
g;or g;in g;part.

, 38 CHAPTER g;2. g; SECOND-ORDER g;DIFFERENTIAL g;EQUATIONS

so g ; these g ; solutions g ; of g ; the g ; associated g ; homogeneous g ; equation g ; are
indepen-dent. g;W ith g;the g;particular g;solution g;yp(x) g;of g;the
g;nonhomogeneous g;equation, g;this g;equation g;has g;general g;solution

4x g; 4 g ; x 1 g; 2
g; g; 1 g;
y(x) g;= g;c1ge
; + g;c2ge
;
4 32 x :

From g;the g;initial g;conditions g;we g;obtain
1
y(0) g;= g;c1 g;+ g;c2 32 g; g; = g;12

and
y0(0) g;= g;4c1 4c2 g;= g;3:
Solve g;these g;to g;obtain g;c1 g;= g;409=64 g;and g;c2 g;= g;361=64 g;to g;obtain g;the g;solution
409 361 4 g ; x g;1 2 g ; g ; 1
y(x) g;= g ; 4x + g; 64g;e 4g;x 32g;:
64 g;e

3. The g;associated g;homogeneous g;equation g;has g;solutions g;e2 g;x g;and
x g;
g;e . g;Their g;Wronskian g;is
2xex
= g;e3xe
W(x)=2xx
2e e
and g;this g;is g;nonzero g;for g;all g;x. g;The g;general g;solution g;of
g;the g;nonhomogeneous g;di g;erential g;equation g;is

2 g ; x g; x g ; 15g:;
y(x) g;= g;c1e + g;c2e + g ;
g
;

2
For g;the g;initial g;value g;problem, g;solve
15
y(0) g;= g;3 g;= c1 g;+ g;c2 g;+ g ; g; 2
and
y0(0) g;= g;1 g;= 2c g ; 1 c2
to g;get g;c1 g;= g;23=2; g;c2 g;= g;22. g;The g;initial g;value g;problem g;has g;solution
23 g;2x x 15
y(x) g;= 2 g;e 22e g ; + 2g:;

4. The g;associated g;homogeneous g;equation g;has g;solutions

y1(x) g;= g;e3x g;cos(2x); g;y2(x) g;= g;e3x g;sin(2x):
The g;W ronskian g;of g;these g;solutions g;is
3x g;
e cos(2x) e3x g;sin(2x) = g;e6x g;6= g;0
3x g; 3x
W g;(x) g;= 3e cos(2x) g ; 2e
g;
sin(2x)

© 2018 Cengage Learning. All Rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

, 2.1. THE g;LINEAR g;SECOND-ORDER g;EQUATION 39

for g;all g;x. g;The g;general g;solution g;of g;the g;nonhomogeneous g;equation g;is

3x 3x 1 g;x
y(x) g;= g;c1e cos(2x) g;+ g;c2e sin(2x) 8 g;e g ; :
g; To g;satisfy g;the g;initial g;conditions, g;it g;is g;required
g; that
1
y(0) g;= g;1 g;= c1
8
and
1

3c1 g;+ g;2c2 8 g ; = g;1:

Solve g;these g;to g;obtain g;c1 g;= g;7=8 g;and g; c2 g;= g;15=8. g;The g;solution g;of
g;the g;initial g;value g;problem g;is

y(x) g;= 7ge; 3x g;cos(2x) g ; + g; 15 g;e3x g ; sin(2x) 1 g;ex:
8 8 8

5. The g;associated g;homogeneous g;equation g;has g;solutions

y1(x) g;= g;ex g;cos(x); g;y2(x) g;= g;ex g;sin(x):
These g;have g;W ronskian
W g;(x) g;= g ; ex g;cos(x) g; ex g;sin(x) ex sin(x) g;+ cos(x) g;= 6= g;0
x 2x
g;e g;e

ex g; cos(x) ex g; sin(x)

so g;these g;solutions g;are g;independent. g;The g;general g;solution g;of
g;the g;nonhomo-geneous g;di g;erential g;equation g;is

5 g; g5
y(x) g;= g;c1ex g;cos(x) g;+ g;c2ex g;sin(x) 2 c2 g ; ; g;
5x 2 :
We
g;need
5
y(0) g;= g;c1 2 g ; = g;6

and

y0(0) g;= g;1 g;= g;c1 g;+ g;c2 5:
Solve g;these g;to g;get g;c1 g; = g;17=2 g;and g;c2 g;= g;5=2 g;to g;get g;the g;solution
g1
; 7 g; x g; g5
; g; x g; g5 5
e sin(x) x2
; g; g; g;
y(x) g;= g ; e cos(x) 5x :
2 2 2 2

6. Suppose g;y1 g;and g;y2 g;are g;solutions g;of g;the g;homogeneous g;equation g;(2.2). g;Then

y100 g;+ g;py10 g;+ g;qy1 g;= g;0

and
© 2018 Cengage Learning. All Rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

, 0
00 g;
y2 + g;py2

© 2018 Cengage Learning. All Rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

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Solution Manual for Advanced Engineering Mathematics, 8th Edition by O'Neil – Complete Chapter 2 Solutions (Second-Order Differential Equations, Wronskian, Euler Equations, Series & Frobenius Methods)

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