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f(x) = T
x(x−l)
l
2
cos
πx
4l
.
f(x) = T
−
1
5
(1 − cos
πx
l
) , 0 ≤ x ≤ l/3
−
9
40l
2
(x − l)
2
l/3 ≤ x ≤ l
.
G = g
x
2
6l
2
, f(x) = T
x(x−l)
l
2
cos
2πx
l
.
G = g
x
2
6l
2
sin
3πvt
l
, F (x) = 0, f (x) =
−7l(1 − cos
πx
l
), 0 ≤ x ≤ l/3
−
63
18l
(x − l)
2
, l/3 ≤ x ≤ l
ψ = g
vt
l
e
πvt/3l
φ = 0
, f(x) = 0 , F (x) = |x −
l
2
| sin
πx
l
.
f(x) = T
0 , 0 ≤ x ≤ l/4
(x−l/4)(l/2−x)
l
2
, l/4 ≤ x ≤ l/2
0 , l/2 ≤ x ≤ l
.
f(x) = T
3
l
2
x
2
, 0 ≤ x ≤ l/8
3
49l
2
(x − l)
2
, l/8 ≤ x ≤ l
.
G = g
x
l
+ e
−
x
l
, f(x) = T
0, 0 ≤ x ≤ l/8
(x−
l
8
)(
l
2
−x)
l
2
, l/8 ≤ x ≤ l/2
0, l/2 ≤ x ≤ l
G = g
x
l
+ 3
2
cos
πvt
l
, f(x) = 0, F (x) =
1
7l
x
2
, 0 ≤ x ≤ l/2
1
7l
(x − l)
2
, l/2 ≤ x ≤ l
φ = g
vt
l
e
−
πvt
l
− e
πvt
l
ψ = 0
, F (x) = 0, f (x) =
(
x(
l
8
−x)
h
, 0 ≤ x ≤ l/8
0, l/8 ≤ x ≤ l
f(x) = T
x(x−l)
l
2
cos
2πx
l
.
f(x) = T
4(1 − cos
πx
l
) , 0 ≤ x ≤ l/5
25
4l
2
(1 − cos
π
5
)(x − l)
2
, l/5 ≤ x ≤ l
.
G = ge
x/4l
, f(x) = T
x(x−l)
l
2
cos
πx
l
.
G = g
3x
2l
sin
πvt
l
, F (x) = 0, f (x) =
4l(1 − c os
πx
l
), 0 ≤ x ≤ l/6
9
25l
(1 −
√
3
2
)(x − l)
2
,l/6 ≤ x ≤ l
ψ = g
e
−2πvt/l
− e
2πvt/l
φ = 0
, f(x) = 0 , F (x) = (x − l) sin
πx
2l
.
f(x) = T
0 , 0 ≤ x ≤ l/8
(x−l/8)(l/2−x)
l
2
, l/8 ≤ x ≤ l/2
0 , l/2 ≤ x ≤ l
.
f(x) = T
3
l
2
x
2
, 0 ≤ x ≤ l/9
3
64l
2
(x − l)
2
, l/9 ≤ x ≤ l
.
G = g
1 −
4x
l
2
, f(x) = T
0, 0 ≤ x ≤ l/8
(x−
l
8
)(
l
4
−x)
l
2
, l/8 ≤ x ≤ l/4
0, l/4 ≤ x ≤ l
G = g
x
l
+ e
−
x
l
cos
3πvt
l
, f(x) = 0, F (x) =
1
7l
x
2
, 0 ≤ x ≤ l/3
1
28l
(x − l)
2
,l/3 ≤ x ≤ l
φ = g
vt
l
e
2πvt
l
ψ = 0
, F (x) = 0, f(x) =
(
0, 0 ≤ x ≤ l/8
(x−l)(
l
8
−x)
h
, l/8 ≤ x ≤ l
f(x) = T
x(x−l)
l
2
cos
πx
l
.
f(x) =
−7(1 − cos
πx
l
) , 0 ≤ x ≤ l/3
−
63
18l
2
(x − l)
2
, l/3 ≤ x ≤ l
.
G = g
3x
2l
, f(x) = |
x
l
−
1
2
| sin
πx
l
.
G = ge
x
l
cos
πvt
l
, F (x) = 0, f (x) =
−
l
6
(1 − cos
πx
l
), 0 ≤ x ≤ 2l/3
−
9
4l
(x − l)
2
, 2l/3 ≤ x ≤ l
ψ = g
e
−3πvt/l
− e
3πvt/l
φ = 0
, f(x) = 0 , F (x) = x sin
2πx
l
.
f(x) = T
0 , 0 ≤ x ≤ l/8
(x−l/8)(l/4−x)
l
2
, l/8 ≤ x ≤ l/4
0 , l/4 ≤ x ≤ l
.
f(x) = T
1
7l
2
x
2
, 0 ≤ x ≤ l/2
1
7l
2
(x − l)
2
, l/2 ≤ x ≤ l
.
G = g
3x
2
4l
2
, f(x) = T
x(l/8−x)
l
2
, 0 ≤ x ≤ l/8
0 , l/8 ≤ x ≤ l
.
12 15
f (x) = T x(x−l)
πx 0, 0 ≤ x ≤ l/8
1.
l2 1 cos 4l . πx (x−l/8)(l/2−x)
− 5 (1 − cos l ) , 0 ≤ x ≤ l/3 1. f (x) = T l2 , l/8 ≤ x ≤ l/2 .
2. f (x) = T 9 2 . 0, l/2 ≤ x ≤ l
− 40l 2 (x − l) l/3 ≤ x ≤ l 3 2
x2
3. G = g 2 , f (x) = T
x(x−l)
cos 2πx x , 0 ≤ x ≤ l/9
l . l 2
6l l2 2. f (x) = T 3 2 .
−7l(1 − cos πx (x − l) , l/9 ≤ x≤l
l ), 0 ≤ x ≤ l/3
2 64l 2
x 3πvt
4. G = g 2 sin , F (x) = 0, f (x) =
0, l l 0 ≤ x ≤ l/8
63
6l l − 18l (x − l)2 , l/3 ≤ x ≤ l
4x 2 (x− 8 )( 4 −x)
ψ = g vtl eπvt/3l 3. G = g 1 − , f (x) = T , l/8 ≤ x ≤ l/4
l
5. , f (x) = 0 , F (x) = |x − 2l | sin πxl .
l2
φ=0 0, l/4 ≤ x ≤ l
1 2
13 7l x , 0 ≤ x ≤ l/3
x x
−l
cos 3πvt
4. G = g
l +e l , f (x) = 0, F (x) = 1
(x − l)2
,l/3 ≤x≤l
28l
0, 0 ≤ x ≤ l/4 (
(x−l/4)(l/2−x)
2πvt
φ = g vtl e l 0, 0 ≤ x ≤ l/8
1. f (x) = T , l/4 ≤ x ≤ l/2 . 5. , F (x) = 0, f (x) = (x−l)( 8l −x)
l2 ψ=0 h , l/8 ≤ x ≤ l
0, l/2 ≤ x ≤ l
3 2
2. f (x) = T l2 x , 0 ≤ x ≤ l/8
. 16
3 2
49l2 (x − l) , l/8 ≤ x ≤ l
0, 0 ≤ x ≤ l/8
x(x−l) πx
(x− 8l )( 2l −x)
1. f (x) = T l2 cos l .πx
x − xl
3. G=g l +e , f (x) = T l/8 ≤ x ≤ l/2, −7(1 − cos l ) , 0 ≤ x ≤ l/3
l2
2. f (x) = .
0, l/2 ≤ x ≤ l 63
− 18l 2 (x − l)
2
, l/3 ≤ x ≤ l
1 2
2 x , 0 ≤ x ≤ l/2 3. G = g
3x x 1 πx
2l , f (x) = | l − 2 | sin l .
x
cos πvt
4. G = g 7l
l +3 l , f (x) = 0, F (x) = 1
(x − l)2
, l/2 ≤ x ≤ l
7l x πvt − 6l (1 − cos πx l ), 0 ≤ x ≤ 2l/3
4. G = ge l cos , F (x) = 0, f (x) =
πvt πvt
( l 9
vt
φ=g l e − l
−e l x( 8 −x)
, 0 ≤ x ≤ l/8 l − 4l (x − l)2 , 2l/3 ≤ x ≤ l
5. , F (x) = 0, f (x) = h
−3πvt/l 3πvt/l
ψ=0 0, l/8 ≤ x ≤ l ψ=g e −e
5. , f (x) = 0 , F (x) = x sin 2πx l .
φ=0
14
17
x(x−l)
1. f (x) = T l2 cos 2πx
l .
0, 0 ≤ x ≤ l/8
4(1 − cos πx l ) , 0 ≤ x ≤ l/5 1. f (x) = T (x−l/8)(l/4−x)
, l/8 ≤ x ≤ l/4 .
2. f (x) = T 25 π 2 . l2
4l2 (1 − cos 5 )(x − l) , l/5 ≤ x ≤ l
0, l/4 ≤ x ≤ l
3. G = ge
x/4l
, f (x) = T x(x−l) cos πx .
1 2
l2 l
πx 2. f (x) = T 7l2 x , 0 ≤ x ≤ l/2
.
3x πvt 4l(1 − cos
√ l
), 0 ≤ x ≤ l/6 1
(x − l)2
, l/2 ≤ x ≤ l
4. G = g sin , F (x) = 0, f (x) = 7l 2
2l l 9 3 2 x(l/8−x)
25l (1 − 2 )(x − l) ,l/6 ≤ x ≤ l 2 , 0 ≤ x ≤ l/8
−2πvt/l 2πvt/l
3. G = g 3x
4l2 , f (x) = T
l2 .
ψ=g e −e 0, l/8 ≤ x ≤ l
5. , f (x) = 0 , F (x) = (x − l) sin πx 2l .
φ=0
23 24
