147
C H A P T E R 7
Solutions
CHAPTER 1
1. u D
x
2
6t
; u D
2t
x
2
.
2. .a 1/.a C 1/ D 0; 4k
2
2ck C 1 D 0 k
2
C ck 1 D 0.
3. .a 1/.a C 1/ D 0; ck k
2
D 0 cka C ak
2
2a C 1 D 0.
4. a D 4; b D 2.
5. a D b D c:
CHAPTER 2
1. (i) u D
.2x C y/
2
4
and u D
.2x y/
2
4
;
(ii) u D .x 1/e
t
C e
2t
;
(iii) u D
p
x
2
C y
2
and u D
p
x
2
C .2 y/
2
;
(iv) u D e
xCy1
:
2. (i) F D F
p
2
2
x; q; pq y; u 2xp C
2
3
p
3
,
(ii) F D F
1
2q
2
t;
p
q
; pu; 2tuq x
:
3. a; b; and c satisfy a
0
D 6a
2
; b
0
D 6ab; c
0
D 2ac C b
2
:
4 (i) For all k, (ii) k D 2=3, (iii) k D 3=4.
5.
P
T D T ln T C kT;
R
X kX X ln X.
6. Pa D 2ab a;
P
b D a
2
C b
2
:
7. n D 2; F
00
C
4FF
0
D
C
2F
3
D
2
F
2
D
0
D
2
D 0.
CHAPTER 3
1. A D
0
; f D g C 2
.
ln
/
00
;
00
C
.
g.x/ C
/
D 0:
2. f D
n.n C 1/
x
2
; f D
n.n C 1/
cosh
2
x
.
148 7. SOLUTIONS
3. F D
1
a
ln v.
CHAPTER 4
1. (i) p D 3X Y C U Y; q D Y X U Y;.
(ii) p D
2.XP C2Q/
P
; q D
2Xe
Y
P
;
(iii) p D
e
X
P
Q
; q D
e
X
P
Q
2
;
(iv) p D
X
Q
; p D U XP:
2. (i) U D
x
2
p
q
C
c;
(ii) U D u p
2
C c;
(iii) U D x C c;
(iv) A D c
1
p Cc
2
; B D c
1
q C c
3
;
U D c
1
.
u ln p ln q
/
C c
2
x C
1
p
C c
3
y C
1
q
C c
4
;
CHAPTER 5
1. (i) F D F
u;
p
q
;
xp
q
C y
,
(ii) F D F
1
.
x C u; p q
/
; F D F
2
x C y;
p C1
q
,
(iii) F D F
1
.
x q; y C p
/
; F D F
2
.
x C q; y p
/
;
(iv) F D F
1
y C q
1
2
p
2
; F D F
2
.
x C yp; y Cq
/
:
2. P
0
./ D
2
.a C b/
4
F D F
˙p Cabq C b
2
pq
bp.a Cbp/
;
ap.a C bp/x C .abpq C a
2
q p
2
/y C bp.a Cbp/u
abpq C a
2
q p
2
.
3. x D
1
X
; y D
Y
X
Q; u D
U
X
YQ
X
C
1
2
Y
2
X
2
:
4. x D
1
X
; y D Q; u D
YQ U
X
:
CHAPTER 6
1. (i) a
00
D a
2
; b
00
D ab;
(ii) a
xx
C a
yy
D a
2
; b
xx
C b
yy
D ab:
7. SOLUTIONS 149
2. (a) F
00
C 2FF
0
D 0; Q
00
FQ
0
.3F
0
C 2F
2
/Q D 0;
(b) f .u/ D ax Cby Cc; Q D .a
2
C b
2
/
f
00
f
03
for all f .u/; f
0
¤ 0;
(c) f .u/ D .x
2
C y
2
/ C k; Q D 4k
f
00
f
03
4
ff
00
f
02
f
03
for all f .u/; f
0
¤ 0:
3. (a) Q D A.u
t
/ C B.u
x
/; A; B arbitrary;
(b) Q D F .u/
u
2
t
u
2
x
C G.u/ where F
00
C 2FF
0
D 0; G
00
C 2F
0
G D 0:
4. Q.u; p/ D
f
00
p
2
C c
1
f C C.f
0
p/
f
0
where c
1
and C is an arbitrary constant and function.
..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.
Reset