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G = g
4x
l
, f(x) = T |
x
l
−
1
2
| sin
2πx
l
.
G = g
x
3l
cos
3πvt
l
, F (x) = 0, f (x) =
−
l
3
(1 − cos
πx
l
), 0 ≤ x ≤ l/2
−
4
3l
(x − l)
2
, l/2 ≤ x ≤ l
ψ = g
v
2
t
2
l
2
e
2πvt/l
φ = 0
, f(x) = 0 , F (x) =
x(x−l)
h
cos
πx
4l
.
f(x) = T
0 , 0 ≤ x ≤ l/4
(x−l/4)(3l/4−x)
l
2
, l/4 ≤ x ≤ 3l/4
0 , 3l/4 ≤ x ≤ l
.
f(x) = T
(
−
3x
2
l
2
, 0 ≤ x ≤ l/5
−
3(x−l)
2
16l
2
, l/5 ≤ x ≤ l
.
G = g
2x
2
l
2
, f(x) = T
x(3l/4−x)
l
2
, 0 ≤ x ≤ 3l/4
0 , 3l/4 ≤ x ≤ l
.
G = g
1 − e
−
2x
l
sin
πvt
l
, f(x) = 0, F (x) =
3
l
x
2
, 0 ≤ x ≤ l/7
1
12l
(x − l)
2
,l/7 ≤ x ≤ l
φ = g
e
3πvt
l
− 1
ψ = 0
, F (x) = 0, f (x) =
0, 0 ≤ x ≤ l/4
(x−l/4)(l/2−x)
h
,l/4 ≤ x ≤ l/2
0, l/2 ≤ x ≤ l
f(x) = T
|x−
l
2
|
l
sin
2πx
l
.
f(x) = T
−
1
3
(1 − cos
πx
l
) , 0 ≤ x ≤ l/2
−
4
3l
2
(x − l)
2
, l/2 ≤ x ≤ l
.
G = g
e
x/l
+ 1
, f(x) = T
(x−l)
2
l
2
sin
πx
4l
.
G = g
x
2
l
2
sin
πvt
l
, F (x) = 0, f(x) =
−
l
5
(1 − cos
πx
l
), 0 ≤ x ≤ l/3
−
9
40l
(x − l)
2
l/3 ≤ x ≤ l
ψ = g
e
−πvt/l
− e
πvt/l
φ = 0
, f(x) = 0 , F (x) =
x(x−l)
h
cos
2πx
l
.
f(x) = T
x(3l/4−x)
l
2
, 0 ≤ x ≤ 3l/4
0 , 3l/4 ≤ x ≤ l
.
f(x) = T
3
l
2
x
2
, 0 ≤ x ≤ l/6
3
25l
2
(x − 2)
2
l/6 ≤ x ≤ l
.
G = ge
−2x/l
, f(x) = T
0 , 0 ≤ x ≤ 3l/4
(x−l)(3l/4−x)
l
2
, 3l/4 ≤ x ≤ l
.
G = g
3x
l
− e
−
x
l
cos
3πvt
l
, f(x) = 0, F (x) =
3
l
x
2
, 0 ≤ x ≤ l/8
3
49l
(x − l)
2
,l/8 ≤ x ≤ l
φ = g
e
−
πvt
2l
− e
πvt
2l
ψ = 0
, F (x) = 0, f(x) =
0, 0 ≤ x ≤ l/8
(x−
l
8
)(
l
2
−x)
h
,l/8 ≤ x ≤ l/2
0, l/2 ≤ x ≤ l
f(x) = T
(x−l)
2
l
2
sin
πx
4l
.
f(x) =
4(1 − cos
πx
l
) , 0 ≤ x ≤ l/4
64
9l
2
(1 −
√
2
2
)(x − l)
2
l/4 ≤ x ≤ l
.
G = g
3x
2
l
2
−
x
l
, f(x) = T
x(x−l)
l
2
cos
πx
4l
.
G = g
x
2
l
2
sin
2πvt
l
, F (x) = 0, f(x) =
4l(1 − cos
πx
l
), 0 ≤ x ≤ l/5
25
4l
(1 − cos
π
5
)(x − l)
2
,l/5 ≤ x ≤ l
ψ = g
vt
l
e
πvt/2l
φ = 0
, f(x) = 0 , F (x) =
x(x−l)
h
cos
πx
l
.
f(x) = T
0 , 0 ≤ x ≤ 3l/4
(x−l)(3l/4−x)
l
2
, 3l/4 ≤ x ≤ l
.
f(x) = T
3
l
2
x
2
, 0 ≤ x ≤ l/7
1
12l
2
(x − l)
2
, l/7 ≤ x ≤ l
.
G = g
x
3l
, 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
.
G = g
x
2
l
2
+ e
x
l
cos
2πvt
l
, f(x) = 0, F (x) =
3
l
x
2
, 0 ≤ x ≤ l/9
3
64l
(x − l)
2
,l/9 ≤ x ≤ l
φ = g
v
2
t
2
l
2
e
−
πvt
3l
ψ = 0
, F (x) = 0, f (x) =
0, 0 ≤ x ≤ l/8
(x−
l
8
)(
l
4
−x)
h
, l/8 ≤ x ≤ l/4
0, l/4 ≤ x ≤ l
3 2
3. G = g 4x x 1 2πx
l , f (x) = T | l − 2 | sin l . 2. f (x) = T l2 x , 0 ≤ x ≤ l/6
.
3
− 3l (1 − cos πx − 2)2 l/6 ≤ x ≤ l
25l2 (x
x
G = g 3l cos 3πvt l ), 0 ≤ x ≤ l/2
4.
l , F (x) = 0, f (x) =
4
− 3l (x − l)2 , l/2 ≤ x ≤ l −2x/l 0, 0 ≤ x ≤ 3l/4
3. G = ge , f (x) = T (x−l)(3l/4−x) .
2 2
ψ = g vl2t e2πvt/l x(x−l) l2 , 3l/4 ≤ x ≤ l
5. , f (x) = 0 , F (x) = cos πx
4l .
3 2
lx , 0 ≤ x ≤ l/8
φ=0 h
3x − xl
cos 3πvt
4. G = g
l −e l , f (x) = 0, F (x) = 3 2
(x − l) ,l/8 ≤ x ≤ l
49l
7 πvt πvt
0, 0 ≤ x ≤ l/8
φ = g e− 2l − e 2l
l l
(x− 8 )( 2 −x)
5. , F (x) = 0, f (x) = ,l/8 ≤ x ≤ l/2
0, 0 ≤ x ≤ l/4 ψ=0 h
0, l/2 ≤ x ≤ l
(x−l/4)(3l/4−x)
1. f (x) = T l2 , l/4 ≤ x ≤ 3l/4 .
0, 3l/4 ≤ x ≤ l
( 2
10
− 3x l2 , 2 0 ≤ x ≤ l/5
2. f (x) = T .
− 3(x−l) , l/5 ≤ x ≤ l f (x) = T (x−l)
2
sin πx
4l .
16l2 1.
x(3l/4−x)
, 0 ≤ x ≤ 3l/4 l2
2x2
3. G = g 2 , f (x) = T l2 . 4(1 − cos√ πx l ) , 0 ≤ x ≤ l/4
l 0, 3l/4 ≤ x ≤ l 2. f (x) = 64 2 .
2
(1 − )(x − l) l/4 ≤ x ≤ l
3 2 2 9l2 2
lx , 0 ≤ x ≤ l/7
− 2x
4. G = g 1 − e l sin πvtl , f (x) = 0, F (x) = 1 2 3. G = g
3x x
l2 − l , f (x) = T
x(x−l)
cos πx .
12l (x − l) ,l/7 ≤ x ≤ l l2
4l
4l(1 − cos πx l ), 0 ≤ x ≤ l/5
0, 0 ≤ x ≤ l/4 x2 2πvt
3πvt
φ=g e l −1 4. G = g 2 sin , F (x) = 0, f (x) = 25 π 2
(x−l/4)(l/2−x) l l (1 − cos )(x − l) ,l/5 ≤x≤l
5. , F (x) = 0, f (x) = h ,l/4 ≤ x ≤ l/2 4l 5
ψ=0 vt πvt/2l
0, l/2 ≤ x ≤ l ψ=g le
5. , f (x) = 0 , F (x) = x(x−l) h cos πxl .
φ=0
8
11
|x− 2l | 2πx
1. f (x) = T l sin l .
0, 0 ≤ x ≤ 3l/4
− 31 (1 − cos πx l ) , 0 ≤ x ≤ l/2 1. f (x) = T (x−l)(3l/4−x) .
2. f (x) = T . ,3l/4 ≤ x ≤ l
− 3l42 (x − l)2 , l/2 ≤ x ≤ l 3 2
l2
2 l2 x , 0 ≤ x ≤ l/7
2. f (x) = T .
G = g ex/l + 1 , f (x) = T (x−l) sin πx
3. . 1
l)2 ,
l2 4ll πx 12l2 (x − l/7 ≤ x ≤ l
− 5 (1 − cos l ), 0 ≤ x ≤ l/3
4.
2
G = g xl2 sin πvt , F (x) = 0, f (x) = 9
0, 0 ≤ x ≤ l/4
l − 40l (x − l)2 l/3 ≤ x ≤ l x
3. G = g , f (x) = T
(x−l/4)(l/2−x)
, l/4 ≤ x ≤ l/2 .
−πvt/l πvt/l
3l l2
ψ=g e −e x(x−l)
0, l/2 ≤ x ≤ l
5. , f (x) = 0 , F (x) = h cos 2πx l .
φ=0 3 2
lx , 0 ≤ x ≤ l/9
2 x
x 2πvt
4. G = g
l2 + e cos l , f (x) = 0, F (x) =
l
3 2
64l (x − l) ,l/9 ≤x≤l
9
0, l l 0 ≤ x ≤ l/8
2 2 πvt
x(3l/4−x) φ = g vl2t e− 3l (x− 8 )( 4 −x)
, 0 ≤ x ≤ 3l/4 5. , F (x) = 0, f (x) = , l/8 ≤ x ≤ l/4
1. f (x) = T l2 . ψ=0 h
0, 3l/4 ≤ x ≤ l 0, l/4 ≤ x ≤ l
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