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G = g
2x
2
l
2
+ e
x/l
, f(x) = 0 , F(x) = (x − l) sin
πx
2l
.
G = ge
2x
l
sin
πvt
l
, F (x) = 0, f(x) =
4l(1 − cos
πx
l
), 0 ≤ x ≤ l/7
49
9l
(1 − cos
π
7
)(x − l)
2
,l/7 ≤ x ≤ l
ψ = g
e
−πvt/2l
− e
πvt/2l
φ = 0
, f(x) = 0 , F (x) =
x(x−l)
h
cos
πx
2l
.
F (x) = 0 , f(x) =
x(l/8−x)
h
, 0 ≤ x ≤ l/8
0 , l/8 ≤ x ≤ l
.
f(x) = 0 , F (x) =
1
7l
x
2
, 0 ≤ x ≤ l/3
1
28l
(x − l)
2
, l/3 ≤ x ≤ l
.
G = g
1 − e
−
2x
l
, F (x) = 0, f (x) =
(
0, 0 ≤ x ≤ l/8
(x−l)(
l
8
−x)
h
, l/8 ≤ x ≤ l
G = g
3x
2
l
2
−
x
l
cos
2πvt
l
, f(x) = 0, F (x) =
1
7l
x
2
, 0 ≤ x ≤ 2l/3
4
7l
(x − l)
2
,2l/3 ≤ x ≤ l
φ = g
vt
l
e
πvt
2l
ψ = 0
, F (x) = 0, f(x) =
(
0, 0 ≤ x ≤ 3l/8
(x−l)(x−
3l
8
)
h
, 3l/8 ≤ x ≤ l
f(x) = 0 , F (x) = (x − l) sin
πx
2l
.
F (x) = 0 , f(x) =
−
l
6
(1 − cos
πx
l
) , 0 ≤ x ≤ 2l/3
−
9
4l
(x − l)
2
, 2l/3 ≤ x ≤ l
.
G = g
x
l
−
6x
2
l
2
, f(x) = 0 , F (x) = x sin
2πx
l
.
G = g
1 −
x
l
2
cos
3πvt
l
, F (x) = 0,f(x) =
2l(1 − cos
πx
l
), 0 ≤ x ≤ 2l/3
27
l
(x − l)
2
, 2l/3 ≤ x ≤ l
ψ = g
vt
l
e
−πvt/3l
φ = 0
, f(x) = 0 , F (x) = (x − l) sin
2πx
l
.
F (x) = 0 , f(x) =
0 , 0 ≤ x ≤ l/8
(x−l)(l/8−x)
h
, l/8 ≤ x ≤ l
.
f(x) = 0 , F (x) =
1
7l
x
2
, 0 ≤ x ≤ l/4
1
63l
(x − l)
2
, l/4 ≤ x ≤ l
.
G = g
x
l
− 1
, F (x) = 0 , f(x) =
x(x−3l/8)
h
, 0 ≤ x ≤ 3l/8
0 , 3l/8 ≤ x ≤ l
.
G = g
x
l
− 1
sin
2πvt
l
, f(x) = 0, F (x) =
4l(1 − cos
πx
l
),0 ≤ x ≤ 2l/3
54
l
(x − l)
2
, 2l/3 ≤ x ≤ l
φ = g
e
−
πvt
4l
− e
πvt
4l
ψ = 0
, F (x) = 0, f(x) =
(
0, 0 ≤ x ≤ l/4
(x−l)(
l
4
−x)
h
, l/4 ≤ x ≤ l
f(x) = 0 , F (x) = x sin
2πx
l
.
F (x) = 0 , f(x) =
4l(1 − cos
πx
l
) , 0 ≤ x ≤ l/7
49
9l
(1 − cos
π
7
)(x − l)
2
, l/7 ≤ x ≤ l
.
G = g
x
l
+ 3
2
, f (x) = 0 , F (x) =
x(x−l)
h
cos
πx
2l
.
G = g
3x
2
4l
2
sin
3πvt
l
, F (x) = 0, f(x) =
(
4l(1 − cos
πx
l
), 0 ≤ x ≤ l/2
16(x−l)
2
l
, l/2 ≤ x ≤ l
ψ = g
v
2
t
2
l
2
e
πvt/4l
φ = 0
, f(x) = 0 , F (x) =
x
2
h
sin
πx
l
.
F (x) = 0 , f(x) =
x(x−3l/8)
h
, 0 ≤ x ≤ 3l/8
0 , 3l/8 ≤ x ≤ l
.
f(x) = 0 , F (x) =
1
7l
x
2
, 0 ≤ x ≤ 2l/3
4
7l
(x − l)
2
, 2l/3 ≤ x ≤ l
.
G = g
3x
l
− e
−
x
l
, F (x) = 0, f (x) =
(
0, 0 ≤ x ≤ 3l/8
(x−l)(x−
3l
8
)
h
, 3l/8 ≤ x ≤ l
G = g
2x
l
+ 4
cos
πvt
l
, f(x) = 0, F (x) =
(
−
3x
2
l
, 0 ≤ x ≤ l/3
−
3(x−l)
2
4l
, l/3 ≤ x ≤ l
φ = g
vt
l
e
−
πvt
l
− e
πvt
l
ψ = 0
, F (x) = 0, f (x) =
(
0, 0 ≤ x ≤ l/2
(x−l)(
l
2
−x)
h
,l/2 ≤ x ≤ l
1 2
7l x , 0 ≤ x ≤ l/4
2x2
3. G = g l2 , f (x) = 0 , F (x) = (x − l) sin πx
+ ex/l 2l . 2. f (x) = 0 , F (x) = .
1
πx 63l (x − l)2 ,l/4 ≤ x ≤ l
2x 4l(1 − cos l ), 0 ≤ x ≤ l/7
4. G = ge l sin πvt
x(x−3l/8)
l , F (x) = 0, f (x) = 49 (1 − cos π )(x − l)2 ,l/7 ≤ x ≤ l 3. G = g xl − 1 , F (x) = 0 , f (x) =
, 0 ≤ x ≤ 3l/8
9l 7
h .
0 , 3l/8 ≤ x ≤ l
ψ = g e−πvt/2l − eπvt/2l
5. , f (x) = 0 , F (x) = x(x−l)
h cos πx
2l . 4l(1 − cos πx l ),0 ≤ x ≤ 2l/3
φ=0 4. G = g xl − 1 sin 2πvt
l , f (x) = 0, F (x) = 54 2
πvt πvt
(l (x − l) , 2l/3 ≤ x ≤ l
ÂÀ ÈÀÍÒ 19 φ=g e − 4l
−e 4l 0, 0 ≤ x ≤ l/4
5. , F (x) = 0, f (x) = (x−l)( 4l −x)
x(l/8−x) ψ=0 h , l/4 ≤ x ≤ l
, 0 ≤ x ≤ l/8
1. F (x) = 0 , f (x) = h .
0, l/8 ≤ x ≤ l
1 2 ÂÀ ÈÀÍÒ 22
2. f (x) = 0 , F (x) = 7l x , 0 ≤ x ≤ l/3
.
1 2
28l (x − l) , l/3 ≤ x ≤ l 2πx
( 1. f (x) = 0 , F (x) = x sin l .
0, 0 ≤ x ≤ l/8
4l(1 − cos πx
2x
3. G = g 1 − e− l , F (x) = 0, f (x) = (x−l)( 8l −x) 2. F (x) = 0 , f (x) = l ) , 0 ≤ x ≤ l/7
.
, l/8 ≤ x ≤ l 49
(1 − cos π 2
h 1 2 9l 7 )(x − l) , l/7 ≤ x ≤ l
2
7l x , 0 ≤ x ≤ 2l/3 , f (x) = 0 , F (x) =(x(x−l)
x
cos πx
2
4. G = g 3x x 2πvt 3. G = g +3 2l .
l2 − l cos l , f (x) = 0, F (x) = 4
(x − l)2
,2l/3 ≤ x ≤ l
l h
7l 2 4l(1 − cos πx l ), 0 ≤ x ≤ l/2
4. G = g 3x sin 3πvt
(
l , F (x) = 0, f (x) =
vt πvt 0, 0 ≤ x ≤ 3l/8
φ = g l e 2l 4l2 16(x−l)2
, l/2 ≤ x ≤ l
5. , F (x) = 0, f (x) = (x−l)(x− 3l ) l
ψ=0 8
, 3l/8 ≤ x ≤ l 2 2
h
ψ = g vl2t eπvt/4l x2
5. , f (x) = 0 , F (x) = h sin πx
l .
φ=0
ÂÀ ÈÀÍÒ 20
πx ÂÀ ÈÀÍÒ 23
1. f (x) = 0 , F (x) = (x − l) sin 2l .
− 6l (1 − cos πx
l ) , 0 ≤ x ≤ 2l/3 x(x−3l/8)
2. F (x) = 0 , f (x) = . , 0 ≤ x ≤ 3l/8
9
− 4l (x − l)2 , 2l/3 ≤ x ≤ l 1. F (x) = 0 , f (x) = h .
0, 3l/8 ≤ x ≤ l
x 6x2
, f (x) = 0 , F (x) = x sin 2πx
1 2
3. G = g l − l2 l . 7l x , 0 ≤ x ≤ 2l/3
2. f (x) = 0 , F (x) = 4 2 .
2l(1 − cos πx (x − l) , 2l/3 ≤x≤l
l ), 0 ≤ x ≤ 2l/3
2 7l
4. G = g 1 − xl cos 3πvtl , F (x) = 0,f (x) = 27 (x − l)2 ,
(
2l/3 ≤ x ≤ l − xl
0, 0 ≤ x ≤ 3l/8
l 3. G = g 3x
ψ = g vtl e−πvt/3l l −e , F (x) = 0, f (x) = (x−l)(x− 3l
8 )
5. , f (x) = 0 , F (x) = (x − l) sin 2πx , 3l/8 ≤ x ≤ l
φ=0 l . ( h 2
3x
− l , 0 ≤ x ≤ l/3
4. G = g 2x πvt
l + 4 cos l , f (x) = 0, F (x) = 3(x−l)2
ÂÀ ÈÀÍÒ 21 − , l/3 ≤ x ≤ l
πvt πvt
( 4l
vt
φ=g l e − l
−e l 0, 0 ≤ x ≤ l/2
5. , F (x) = 0, f (x) =
0, 0 ≤ x ≤ l/8 (x−l)( 2l −x)
1. F (x) = 0 , f (x) = (x−l)(l/8−x) . ψ=0 h ,l/2 ≤ x ≤ l
h , l/8 ≤ x ≤ l
41 42
