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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) = 0 , F (x) = |x −
l
2
| sin
2πx
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
.
G = g
e
x/l
+ 1
, f(x) = 0 , F (x) =
(x−l)
2
h
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) = 0 , f(x) =
x(3l/4−x)
h
, 0 ≤ x ≤ 3l/4
0 , 3l/4 ≤ x ≤ l
.
f(x) = 0 , F (x) =
3
l
x
2
, 0 ≤ x ≤ l/6
3
25l
(x − 2)
2
l/6 ≤ x ≤ l
.
G = ge
−2x/l
, F (x) = 0 , f(x) =
0 , 0 ≤ x ≤ 3l/4
(x−l)(3l/4−x)
h
, 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) = 0 , F (x) =
(x−l)
2
h
sin
πx
4l
.
F (x) = 0 , f(x) =
4l(1 − cos
πx
l
) , 0 ≤ x ≤ l/4
64
9l
(1 −
√
2
2
)(x − l)
2
l/4 ≤ x ≤ l
.
G = g
3x
2
l
2
−
x
l
, f(x) = 0 , F (x) =
x(x−l)
h
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) = 0 , f(x) =
0 , 0 ≤ x ≤ 3l/4
(x−l)(3l/4−x)
h
, 3l/4 ≤ x ≤ l
.
f(x) = 0 , F (x) =
3
l
x
2
, 0 ≤ x ≤ l/7
1
12l
(x − l)
2
, l/7 ≤ x ≤ l
.
G = g
x
3l
, 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
.
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
f(x) = 0 , F (x) =
x(x−l)
h
cos
πx
4l
.
F (x) = 0 , f(x) =
−
l
5
(1 − cos
πx
l
) , 0 ≤ x ≤ l/3
−
9
40l
(x − l)
2
l/3 ≤ x ≤ l
.
G = g
x
2
6l
2
, f(x) = 0 , F (x) =
x(x−l)
h
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
.
3 2 πx
lx , 0 ≤ x ≤ l/7 4l(1 − cos ), 0 ≤ x ≤ l/4
2x
4. G = g 1 − e− l sin πvt
l , f (x) = 0, F (x) = 1 2 2. F (x) = 0 , f (x) = √ l .
12l (x − l) ,l/7 ≤ x ≤ l 64
(1 − 2
)(x − l)2 l/4 ≤ x ≤ l
9l 2
0, 0 ≤ x ≤ l/4
3x2
, f (x) = 0 , F (x) = x(x−l)
3πvt
x
φ=g e l −1
(x−l/4)(l/2−x)
3. G = g l2 − l h cos πx
4l .
5. , F (x) = 0, f (x) = h ,l/4 ≤ x ≤ l/2 πx
ψ=0 2 4l(1 − cos l ), 0 ≤ x ≤ l/5
0, l/2 ≤ x ≤ l 4. G = g xl2 sin 2πvt
l , F (x) = 0, f (x) = 25 (1 − cos π )(x − l)2 ,l/5 ≤ x ≤ l
4l 5
ψ = g vtl eπvt/2l x(x−l)
ÂÀ ÈÀÍÒ 8 5. , f (x) = 0 , F (x) = h cos πx l .
φ=0
1. f (x) = 0 , F (x) = |x − 2l | sin 2πx
l . ÂÀ ÈÀÍÒ 11
− 3 (1 − cos πx
l
2. F (x) = 0 , f (x) = l ) , 0 ≤ x ≤ l/2
.
4
(x − l)2 ,
− 3l l/2 ≤ x ≤ l 0, 0 ≤ x ≤ 3l/4
2 1. F (x) = 0 , f (x) = (x−l)(3l/4−x) .
3. G = g ex/l + 1 , f (x) = 0 , F (x) = (x−l) sin πx , 3l/4 ≤ x ≤ l
l h 4l . 3 2
h
lx , 0 ≤ x ≤ l/7
πx
2 − (1 − cos l ), 0 ≤ x ≤ l/3 2. f (x) = 0 , F (x) = .
4. G = g xl2 sin πvt , F (x) = 0, f (x) = 5 1
l)2 ,
12l (x −l/7 ≤ x ≤ l
l 9 2
− 40l (x − l) l/3 ≤ x ≤ l
πvt/l 0, 0 ≤ x ≤ l/4
−πvt/l
ψ =g e −e
5. , f (x) = 0 , F (x) = x(x−l)
h cos 2πx
l . x
3. G = g 3l , F (x) = 0 , f (x) = (x−l/4)(l/2−x)
, l/4 ≤ x ≤ l/2 .
φ =0 h
0, l/2 ≤ x ≤ l
3 2
ÂÀ ÈÀÍÒ 9 x , 0 ≤ x ≤ l/9
2 x
4. G = g xl2 + e l cos 2πvt l , f (x) = 0, F (x) =
l
3 2
64l (x − l) ,l/9 ≤x≤l
x(3l/4−x)
, 0 ≤ x ≤ 3l/4 0, l l
0 ≤ x ≤ l/8
1. F (x) = 0 , f (x) = h . 2 2 πvt
0, 3l/4 ≤ x ≤ l φ = g vl2t e− 3l (x− 8 )( 4 −x)
5. , F (x) = 0, f (x) = , l/8 ≤ x ≤ l/4
ψ=0 h
3 2
lx , 0 ≤ x ≤ l/6
2. f (x) = 0 , F (x) = 3 2 . 0, l/4 ≤ x ≤ l
25l (x − 2)
l/6 ≤ x ≤ l
0, 0 ≤ x ≤ 3l/4 ÂÀ ÈÀÍÒ 12
3. G = ge−2x/l , F (x) = 0 , f (x) = (x−l)(3l/4−x) .
h , 3l/4 ≤ x ≤ l
3 2 x(x−l)
− xl lx , 0 ≤ x ≤ l/8 1. f (x) = 0 , F (x) = cos πx
4l .
4. G = g 3x 3πvt
− e cos , f (x) = 0, F (x) = h
3 2
l l (x − l) ,l/8 ≤x≤l − 5 (1 − cos πx
l
49l 2. F (x) = 0 , f (x) = l ) , 0≤x ≤ l/3
.
9
πvt πvt
0, l l 0 ≤ x ≤ l/8 − 40l (x − l)2 l/3 ≤ x≤l
φ = g e− 2l − e 2l
, F (x) = 0, f (x) = (x− 8 )(
2
5.
h
2 −x)
,l/8 ≤ x ≤ l/2 x
3. G = g 6l 2 , f (x) = 0 , F (x) = x(x−l) 2πx
h cos l .
ψ=0 0, l/2 ≤ x ≤ l −7l(1 − cos πx
l ), 0 ≤ x ≤ l/3
2
x 3πvt
4. G = g 6l 2 sin l , F (x) = 0, f (x) = 63
− 18l (x − l)2 , l/3 ≤ x ≤ l
ÂÀ ÈÀÍÒ 10 ψ = g vtl eπvt/3l
5. , f (x) = 0 , F (x) = |x − 2l | sin πx
l .
φ=0
(x−l)2
1. f (x) = 0 , F (x) = h sin πx
4l .
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