E
j
e
r
c
i
c
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e
s
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l
t
H
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g
d
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L
1
1
E
1
1
.
c
1
u
E
a
c
c
u
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d
s
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r
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r
i
y
r
i
a
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u
a
b
c
s
c
b
A
n
e
e
m
3
e
l
e
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d
l
l
a
o
2
e
i
F
0
1
c
i
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r
e
b
n
e
e
s
s
p
l
e
r
s
i
m
a
e
e
s
dy + 2y 2y = 0 dx e
f
i
n
i
m
o
s
e
l
f
a
c
p( p(x) = 2 f
a
c
t
m
o
u
l
r
t
i
i
p
n
l
t
i
e
c
g
a
r
a
m
n
o
t
s
e
l
:
o
´
2dx
e
a
t
e
c
r
i
n
t
a
c
i
g
r
a
n
t
e
.
e2x
=
u
e
o
n
p
o
r
e
l
f
a
c
t
o
r
i
n
t
e
g
r
a
n
e2x ddxy + 2e 2e2x = 0 e
l
l
a
d
o
i
z
q
d 2x dx [e y ] s
e
p
a
r
a
m
u
i
e
r
d
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d
e
l
a
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c
u
a
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d
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a
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a
b
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t
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g
´
r
a
m
o
s
.
d 2x dx [e y ]
=0
´
dx + c
y = ce−2x .
dy = 3y 3y dx f
o
r
m
a
l
i
n
e
a
l
.
dy dx
− 3y = 0 p( p(x) = −3
´
F
m
a
c
u
t
l
o
t
r
i
p
i
l
n
i
t
c
e2x y = c
2
:
=0 o
e
a
g
m
r
o
a
n
s
t
p
e
o
:
r
e −3dx e−3x
=
f
a
c
t
o
r
i
n
t
e
g
r
a
n
t
e
.
1
i
f
e
r
e
n
c
i
a
.
D
d
d
u
o
l
1
s
d
c
t
e
.
t
r
a
o
s
.
r
d
e
n
l
e
s
e−3x ddxy
− 3e−3xy = 0
´ dy −3x ´ [ e y = 0 dx + c dx e−3x y = c y = ce3x 3
.
3 p
a
s
a
m
o
s
l
a
e
c
u
a
c
i
o
n
a
l
a
f
o
r
m
dy + 12y 12y = 4 dx a
l
dy dx
i
n
e
a
l
.
+ 4y 4y =
4 3
p( p(x) = 4 ´
F
a
c
t
o
r
i
n
t
e
g
r
a
n
t
e
:
e
4dx
e4x
=
e4x ddxy + 4e 4e4x y = 43 e4x
´
d 4x dx [e y ]
=
´
e4x dx + c
e4x y = 14 e4x + c y= 4
1 4
+ ce−4x
.
y = 2y 2 y + x2 + 5
f
o
r
m
a
l
i
n
e
a
l
y
− 2y = x2 + 5
´
F
a
c
t
o
r
i
n
t
e
g
r
a
n
t
e
:
e −2dx = e−2x e−2x y
´
5e−2x − 2e−2xy = e−2xx2 + 5e
d 2x y] dx [e
−
=
´ −2x 2 ´ e x + 5 e−2x + c
(2x2 + 2x 2x + 1) + C − 52 e−2x − 14 e−2x(2x y = − x2 − x2 − 14 + 52 + ce2x
e−2x y =
2
5
.
ydx
4(x + y 6 )dy )dy = 0 − 4(x
ydx = 4(x 4( x + y6 )dy )dy dx dy
=
4(x+y6 ) y
dx dy
;
2
=
4x y
+
4y6 y
e−3x ddxy
− 3e−3xy = 0
´ dy −3x ´ [ e y = 0 dx + c dx e−3x y = c y = ce3x 3
.
3 p
a
s
a
m
o
s
l
a
e
c
u
a
c
i
o
n
a
l
a
f
o
r
m
dy + 12y 12y = 4 dx a
l
dy dx
i
n
e
a
l
.
+ 4y 4y =
4 3
p( p(x) = 4 ´
F
a
c
t
o
r
i
n
t
e
g
r
a
n
t
e
:
e
4dx
e4x
=
e4x ddxy + 4e 4e4x y = 43 e4x
´
d 4x dx [e y ]
=
´
e4x dx + c
e4x y = 14 e4x + c y= 4
1 4
+ ce−4x
.
y = 2y 2 y + x2 + 5
f
o
r
m
a
l
i
n
e
a
l
y
− 2y = x2 + 5
´
F
a
c
t
o
r
i
n
t
e
g
r
a
n
t
e
:
e −2dx = e−2x e−2x y
´
5e−2x − 2e−2xy = e−2xx2 + 5e
d 2x y] dx [e
−
=
´ −2x 2 ´ e x + 5 e−2x + c
(2x2 + 2x 2x + 1) + C − 52 e−2x − 14 e−2x(2x y = − x2 − x2 − 14 + 52 + ce2x
e−2x y =
2
5
.
ydx
4(x + y 6 )dy )dy = 0 − 4(x
ydx = 4(x 4( x + y6 )dy )dy dx dy
=
4(x+y6 ) y
dx dy
;
2
=
4x y
+
4y6 y
d
e
n
i
m
o
s
l
a
f
o
r
m
a
l
i
n
e
a
l
.
dx dy
F
a
c
t
o
r
i
n
t
e
g
r
a
n
t
e
:
´
e−4
1 y dy
− 4yx = 4y5 4
log(y) e−4 log( elog(y )
−
;
;
1 dx y4 dy
y −4 =
− y1 4yx = y1 4y5 4
4
d 1 dy [ y 4 x] d 1 dy [ y4 x]
´
;
1 y4 x
= 4y
´
=4
ydy
= 2y2 + C
x = 2y6 + cy4 6
.
xy + y = ex
ex x
y + x1 y =
F
a
c
t
o
r
i
n
t
e
g
r
a
n
t
e
:
´
e
1 x dx
= elog x = x xex x
xy + xx y =
d xy] dx [xy] I
n
t
e
g
r
a
m
o
s
:
= ex
´
d xy] dx [xy]
=
ex dx + c
´
xy = ex + c y = ex x−1 + cx−1 7
.
x dy dx h
a
c
e
m
o
s
l
a
s
u
s
t
i
t
u
c
i
o
n
:
dy 2 +y = 2 dx y
+
y x
u = y 1−n
d
2 xy2
= o
n
d
e
.
.
.
(
1
n=
)
−2
u = y1−(−2) = y 3 u1/3 = y ;
D
e
r
i
v
a
m
o
s
e
s
t
a
u
l
t
i
m
a
.
1 2/3 du 3u dx
−
3
=
dy dx
1 y4
S
u
s
t
i
t
u
i
m
o
s
e
n
l
a
e
c
u
a
c
i
o
n
d
i
f
e
r
e
n
1 2/3 du 3u dx
−
A
c
o
m
o
d
a
m
o
s
a
l
a
f
o
r
m
a
l
i
n
e
a
l
,
s
t
a
e
s
u
n
a
e
c
u
a
c
i
o
n
l
i
n
u
l
t
i
p
l
i
c
a
m
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s
p
o
r
f
a
c
t
a
o
l
.
D
m
r
i
n
e
a
l
1
.
u1/3 x
u
l
t
i
p
l
i
c
a
n
i
m
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s
e
n
d
o
t
o
d
a
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a
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c
u
a
l
f
a
c
t
o
r
i
n
t
e
g
r
a
n
t
= e3log x = elog x = x3
t
e
g
r
a
n
t
e
.
x3 du 3x3 ux = x3 x6 dx + 3x d 3 dx [x u] i
n
t
e
g
r
a
m
o
s
.
= 6x2
d 3 dx [x u]
´
´
=6
x2 + c
x3 u = 2x3 + c u = 2 + cx−3 S
u
s
t
i
t
u
i
m
o
s
u = y3
y 3 = 2 + cx−3 8
dy y 1/2 dx + y 3/2 = 1
.
;
dy dx
y 3/2 y 1/2
+
u = y 1−n n =
n
d
i
c
i
o
n
y (0) = 4
u = y 1−(−1/2) = y 3/2
−1/2
;
o
dy + y = y −1/2 ↔ dx
1 y 1 /2
=
c
;
u2/3 = y 2 1/3 du dx 3u
−
S
u
s
t
i
t
u
i
m
o
s
.
+ u2/3 = (u ( u2/3 )−1/2
−
u
l
t
i
p
l
i
c
a
m
o
s
l
a
e
c
u
a
c
i
o
n
p
o
r
2 1/3 3u
+ 32 u =
du dx L
F
a
a
e
c
c
t
o
u
a
r
c
i
n
i
o
t
n
e
s
g
dy dx
2 1/3 du dx 3u M
=
r
e
a
r
n
e
t
d
e
u
:
j
e
o
3 2
a
´
u
dx
n
a
l
=e
i
n
e
a
l
.
3 2x
4
c
i
o
6 x
+ 3 ux =
2(u1/3 )2 x
=
3
1 x dx
´
e3 M
e
i
+
du dx E
c
3 2
e
.
n
p
o
r
1 2/3 3u
.
3
3
3
3 3 2x u = e2x e 2 x du dx + e 2 2 3 d 2 x u] dx [e 3 d 2 x u] dx [e
´
3
= 32 e 2 x 3 2x
´ 3
=
2e
3
dx + c
3
e 2 xu = e 2 x + c 3
u = 1 + ce− 2 x S
u
s
t
i
t
u
i
m
o
s
u = y3/2
3
y 3/2 = 1 + ce− 2 x A
h
o
r
a
a
p
l
i
c
a
m
o
s
l
a
s
c
o
n
d
i
c
i
o
n
e
s
i
n
S
o
l
u
c
i
o
n
g
e
n
e
r
a
l
.
i
c
i
a
l
e
s
.
y (0) = 4
3 = 1 + ce− 2 0
43/2
8 1=c c=7
−
S
u
s
t
u
t
u
i
m
o
s
e
l
v
a
l
o
r
d
e
c
e
n
l
a
e
c
u
a
c
i
o
n
g
e
n
e
r
a
l
.
3
y 3/2 = 1 + 7e 7e− 2 x 9
.
y +
u = y 1−n e
n
t
o
n
c
e
s
;
d
:
o
n
d
e
S
o
l
u
c
i
o
n
p
a
r
t
i
c
2 y= x
−2xy2
n =2
1 2
u = y − u = y −1 u−1 = y ;
;
−u−2 ddux = ddxy s
u
s
t
i
t
u
i
m
o
s
e
n
l
a
e
c
u
a
c
i
o
n
.
−u−2 ddux + x2 u−1 = −2x(u−1)2 m
u
l
t
i
p
l
i
c
a
m
o
s
p
o
r
−u2
du dx e
s
o
t
b
a
t
e
e
s
n
u
e
m
n
a
o
e
s
c
e
l
u
f
a
a
c
c
i
t
o
o
n
r
l
i
n
i
n
t
e
e
a
g
l
r
c
a
e−2
o
n
n
t
´
e
.
2x − x2 u = 2x p( p(x) = − x2
1 x dx
x−2 ddux
= elog x
−
n
t
e
g
r
a
m
o
s
.
= x−2
− x−2 x2 u = x−22x
d 2 dx [x u] i
2
−
5
= 2x−1
u
l
a
r
.
d 2 dx [x u]
−
´
=
´
2x−1 dx + c
x−2 u = 2 log x + c u = 2x2 log x + cx2 s
u
s
y
t
l
i
t
a
u
s
i
o
m
l
o
u
c
s
i
u = y−1
ó
n
e
s
e
n
t
o
n
c
e
s
:
1 2x2 log x+cx2
y= 1
0
,
y + xy = xy −1/2
s
e
a
.
n=
−1/2 u = y 1−n u = y 1−(−1/2) u = y 3/2 y = u2/3 ;
;
dy dx s
u
s
t
i
t
u
i
m
o
s
e
n
l
a
e
c
u
a
c
i
o
n
.
+ xu2/3 = x(u2/3 )−1/2
−
u
l
t
i
p
l
i
c
a
du dx F
a
c
t
o
r
i
n
m
o
s
p
o
r
2 1/3 3u
+ 32 xu = 32 x t
e
g
r
a
n
t
e
:
= 23 u−1/3
2 1/3 3u
m
;
q
u
e
e
s
u
n
a
e
c
u
a
c
i
o
n
l
3
3
e4x
´
2
du dx
´
xdx 3
2
+ e4x
3
3 2 xu
t
u
i
m
o
s
l
3 2x
= 32 xe 4 x dx + c
2
d 34 x2 u dx e
=
3
3 2
2
´ 3
2
3
xe 4 x dx + c 2
3
i
2
= e4x
2
u = 1 + ce− 4 x t
a
2
3
d 34 x2 u dx e
3
s
e
= e4x
e4x u = e4x + c
u
n
e2
s
i
u = y3/2 3
y 3/2 = 1 + ce− 4 x
6
2
c
o
n
p(x) = 32 x
1
.
2
E
c
u
a
c
i
o
n
e
s
e
x
a
c
t
a
s
1.(2x M (x, y) = 2x C
o
∂M ∂y
m
p
r
o
b
a
m
y
r
n
A
u
c
i
b
i
h
n
o
t
o
s
q
u
e
l
a
e
c
u
a
c
g
u
r
a
a
l
t
e
s
o
,
m
p
a
o
r
m
l
o
o
s
t
u
a
n
n
a
t
f
o
u
l
i
n
a
o
c
e
i
o
n
c
n
e
g
r
a
m
o
s
r
e
s
p
e
c
t
o
a
x
,
y
l
´ ∂M
s
e
a
e
x
=0
u
a
c
i
o
a
c
t
a
,
∂N ∂x
;
n
a
c
o
n
s
t
e
´
a
s
t
a
f
u
n
c
i
o
n
l
a
d
e
r
i
v
a
m
o
s
c
o
n
r
e
g
e
a
s
t
e
x
a
c
t
a
s
.
o
e
s
s
i
s
e
c
u
m
p
l
e
e
x
a
u
a
l
a
m
o
s
c
o
n
N
(
x
,
y
)
c
t
a
.
n
t
e
t
e
s
d
e
i
n
t
e
e
r
a
c
i
o
n
s
e
r
a
u
n
a
− ´ dx + g(y) .
p
g
c
t
o
d
e
y
.
.
.
(
1
)
= g (y)
g
r
a
m
o
s
r
e
s
p
e
c
t
o
a
y
g(y) = 32 y2 + y + c u
s
t
i
t
u
i
m
o
s
l
a
f
u
n
c
i
o
n
e
n
(
1
)
.
s
2
.
t
a
e
s
u
n
a
s
o
l
u
c
i
o
n
e
n
f
o
r
m
− x + 32 y2 + y = c
x2 e
c
o
n
d
i
c
i
o
n
−1
´ ´ ´ g (y) = 3 ydy + dy + c
s
a
g (y) = 3y + 1 i
l
=0
− x + g(y)
∂f ∂y
i
n
xdx
f (x, y) = x2 s
s
f x (x, y) = M (x, y)
=2
∂x
E
e
− 1; N (x, y) = 3y + 1
f x (x, y) = 2x i
l
− 1)dx + (3y + 1)dy = 0
∂M ∂y o
d
∂N ∂x
=
s
e
a
i
m
p
l
i
c
i
(seny
t
a
d
e
l
a
e
c
u
a
c
i
o
n
.
− ysenx)dx + (cosx + xcosy − y)dy = 0 M (x, y) = seny − ysenx N (x, y) = cosx + xcosy − y ∂M − senx ∂y = cosy ∂N ∂x = −senx + cosy ;
7
f
u
n
c
i
o
n
g(y)
∂M ∂y t
i
o
m
n
a
t
∂N ∂x
=
e
m
g
o
r
a
s
p
r
l
o
t
a
n
t
o
e
s
u
f x (x, y) = seny
m
o
o
s
c
o
n
r
e
s
p
e
c
´
t
o
n
a
e
c
u
e
r
i
v
a
m
o
s
e
s
t
a
e
c
u
a
c
i
o
n
e
x
a
x
´
i
o
n
t
a
.
f x (x, y)dx = (seny
c
c
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f (x, y) = xseny + ycosx n
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− 12 y2 = c
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(3x2 y + ey )dx =
−(x3 + xey − 2y)dy
M (x, y) = 3x2 y + ey N (x, y) = x3 + xey
− 2y
;
M y (x, y) = 3x2 + ey N x (x, y) = 3x2 + ey M y (x, y) = N x (x, y) f x (x, y) I
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f (x, y) = (3x2 y + ey )dx
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x3 y + xey 4
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− 2y2)dx + (3x2 − 4xy)dy = 0 M y (x, y) = 6x − 4y N x (x, y) = 6x − 4y (6xy
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− 4xy + g(y)
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g (y) = 0
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3x2 y 5
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(2y c
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− 2xy3 + 4x + 6)dx
− 3x2y3 + 2x2 + 6x + g(y) .
9
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(
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− x2y3 + 2x2 + 6x − y = c y(−1) = 0 2(−1)2 + 6(−1) = c c = −4
2xy
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− x2y3 + 2x2 + 6x − y = −4
2xy 6
−y + c
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( xy sin x + 2y cos x)dx + 2x cos xdy = 0;
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µ(x, y) = xy
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xy( xy sin x + 2y cos x)dx + xy(2x cos x)dy = 0
−
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−2yx2 sin x + 4xy cos x N X (x, y) = 4xy cos x − 2x2 y sin x
M y (x, y) =
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f x (x, y) = i
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x
:
−x2y2 sin x + 2xy2 cos x
f (x, y) = ( x2 y 2 sin x + 2xy 2 cos x)dx
´
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f (x, y) = x2 y 2 cos x + g(y) .
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f y (x, y) = 2x2 y cos x + g (y)
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f (x, y) = x2 y 2 cos x + c
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:
´
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d
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´ 1
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y = x log x
− x + c1x´ + c2 ´ ´ y = x log xdx − xdx + c1 xdx + c2 dx + c3 y = x2 (log x − 12 ) − 12 x2 + c1 12 x2 + c2 x + c3 y = x + sin x
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´
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sin xdx + c1
− cos x +´ c1 ´ x2 dx − cos xdx + c1 dx + c2 − sin x + c1x + c2 y = x sin x, y(0) = 0 y (0) = 0 y (0) = 2
.
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:
x sin xdx + c!
y = sin x
− x cos x´ + c1 ´ y = sin xdx − x cos xdx + c1 dx + c2 y = − cos x − (cos x + x sin x) + c1 x + c2 ´ ´ ´ ´ ´ ´ y = − cos xdx − cos xdx − x sin xdx + c1 xdx + c2 dx + c3 y = − sin x − sin x − (−x cos x + sin x) + 12 c1 x2 + c2 x + c3 y = −3sin x + x cos x + 12 c1 x2 + c2 x + c3
´
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2
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xy + y = 0
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D
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d
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dy dx
p(x) =
:
dp dx
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=
xp + p = 0
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− p1 d p ´ 1 ´ 1 dx = − x p d p + c1 log x = − log p + log c1 =
log x = log( c p1 ) A
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´ 1
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1 c1
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´
1 c1 y
log x =
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y = c1 log x + c2 v
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dp dx
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(x
x 1 x 1 p
− x−1 1 p = 0
− −
c
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− x−1 1 p = 0
p n
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− x−1 1 p = 0
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log( p) = log(x
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a
c
i
dy dx
e
n
d
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dy dx
p =
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n
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g
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d
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:
´
dy = c1 (x
y = c1 12 x2 3
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3
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− 2y = 0
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m2 + m
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c
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y = emx
y1 = e−2x y2 = ex y(x) = c1 e−2x + c2 ex 2
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y E
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(m
c
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− 2y + y = 0 m2 − 2m + 1 = 0 d
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m1,2 = 1 s
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l
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c
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n
y = emx
y1 = ex y2 = y1
´ e´ p(y)dy
y2 = ex
´ e2x
y12
dx
e2x dx
y2 = ex x s
o
l
u
c
i
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n
.
y(x) = c1 ex + c2 xex 3
.
4y
− 8y + 5y = 0 1
4
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4m2
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m1,2 = 1 o
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c
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− 8m√ + 5 = 0
m1,2 = 8±
s
a
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64 80 8 1 2i
−
±
1
1
y = c1 ex ei 2 x + c2 ex e−i 2 x 1
1
y = ex (c1 ei 2 x + c2 e−i 2 x ) y = ex (c1 cos 12 x + c2 sen 12 x) 4
3y
.
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c
u
− 2y − 8y = 0
a
c
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c
a
r
a
c
t
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r
i
s
t
i
c
a
:
3m2
− 2y − 8 = 0 (3m + 4)(m − 2) m1 = 2
− 43
m2 = S
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c
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p
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a
d
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a
f
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y = emx
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c
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:
4 3x
4
y(x) = c1 e2x + c2 e 3 x 5
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.
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a
c
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m5
− 10m3 + 9m = 0 m(m4 − 10m2 + 9) = 0 m1 = 0 (m2 − 9)(m2 − 1) m2,3 = ±3 m4,5 = ±1 E
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y1 = e0 = 1 y2 = e3x y3 = e−3x y4 = ex y5 = e−x S
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y(x) = c1 + c2 e3x + c3 e−3x + c4 ex + c5 ex 6
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m2 + 4m + 3 = 0
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−2 ± 3i n
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y(x) = e−2x (c1 cos3x + c2 sin3x) y(x) = e−2x (−3c1 sin3x + 3c2 cos3x) − 2e−2x (c1 cos3x + c2 sin3x) R
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y(0) = 2
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2 = e0 (c1 cos 0 + c2 sin 0) 2 = c1 P
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1 3
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− 7m2 − 18 = 0
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y + 3y + 2y = 6
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yh = y + 3y + 2y = 0
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m2 + 3m + 2 = 0 (m
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0 + 3(0) + 2A = 6 A=3 E
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y(x) = yh + y p
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y + y = sin x
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m2 + 1 = 0
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α=0
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y p = Ax sin x + Bx cos x y p = A sin x + Ax cos x + B cos x
− Bx sin x y p = A cos x + A cos x − Ax sin x − B sin x − Bx cos x − B sin x = 2A cos x − 2B sin x − Ax sin x − Bx cos x S
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2A cos x
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A=0 s
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− 10y + 25y = 30x + 3
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− 10m + 25 = 0
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30x + 3
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−10(A) + 25(Ax + B) = 30x + 3 25A = 30
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3 5
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1 4y + y + y = 0 1 2 4m + m + 1 = 0
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− 2x
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1 2 4 (2A) + 2Ax + B + Ax + Bx + C = 1 2 2 2 A + B + Ax + 2Ax + Bx + C = x
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f (x) =
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−48x2e3x
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m2 + 3 = 0
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y p = e3x (Ax2 + Bx + C )
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−48x2e3x
y p = 3e3x (Ax2 + Bx + C ) + e3x (2Ax + B)
y p = 9e3x (Ax2 + Bx + C ) + 3e3x (2Ax + B) + 3e3x (2Ax + B) + 3x 3x 2 3x 3x e (2A) = 9e (Ax + Bx + C ) + 3e (4Ax + 2B) + e (2A)
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3x
9e (Ax + Bx + C ) + 3e (4Ax + 2B) + e3x (2A) + 9e3x (Ax2 + Bx + C ) + 3e3x (2Ax + B) = 48x2 e3x
−
9e3x Ax2 + 9e3x Bx + 9e3x C + 12e3x Ax + 6e3x B + 2e3x A + 9e3x Ax2 + 9e3x Bx + 9e3x C + 6e3x Ax + 3e3x B = 48x2 e3x 9A + 9A = 18A = A=
−48
−
−48
− 83
B =0
C = 0
− y = −3 y m2 − m = 0 m(m − 1) = 0 6
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y
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yh = c1 e0x + c2 ex = c1 + c2 ex E
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y p = Ax
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A=3
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yh = y
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− 6y = 0
m1,2 = 0 m3 = 6 yh = c1 + c2 x + c3 e6x L
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y p1 = Ax2
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Bsenx
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6 37 1 37 1 3 2x
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6 1 37 cosx + 37 senx c1 + c2 x + c3 e6x 14 x2
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y + 2y + y = senx + 3cos2x
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y p = Acosx + Bsenx + Ccos2x + Dsen2x y p =
−Asenx + Bcosx − 2Csen2x + 2Dcos2x
y = −Acosx − Bsenx − 4Ccos2x − 4Dsen2x p
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−Acosx−Bsenx−4Ccos2x−4Dsen2x−2Asenx+2Bcosx−4Csen2x+ 4Dcos2x + Acosx + Bsenx + Ccos2x + Dsen2x = senx + 3cos2x
−3Ccos2x − 3Dsen2x − 2Asenx + 2Bcosx − 4Csen2x + 4Dcos2x = senx + 3Cos2x
−3C + 4D = 3 −3D − 4C = 0 .
.
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(
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.
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(
2
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9 25 12 25
C =
D=
−2A = 1
A=
;
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− 12
;
− 12 cosx + 259 cos2x + 1225 sen2x
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2
1
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5
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y + y = sec x
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m2 + 1 = 0 m2 = m1,2 = m1,2 = α
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yh = c1 cosx + c2 senx A
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y1 = cosx
y2 = senx
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r
i
v
a
m
o
s
.
y1 = cosx y2 = senx y1 = A
c
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i
n
u
a
c
y1 y1
W =
i
o
n
c
y2 = y2
a
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c
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l
a
m
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s
e
l
−senx y2 = cosx W
r
o
n
s
n
o
:
−
− [(senx)(−senx)] = −
y1 y1
W 1 W
a
cosx senx = [(cosx)(cosx)] senx cosx cos2 x + sen2 x = 1
−
u1 =
i
0 y2 0 senx = = [(0)(cosx)] [(senx)(secx)] = f (x) y2 secx cosx senxsecx = senx cosx = tanx
W 1 =
W 2 =
k
0 = f (x)
= −tanx = 1
−
−
cosx 0 = [(cosx)(secx) senx secx cosxsecx = cosx cosx = 1
−
−tanx u2 =
u1 =
;
W 2 W
=
1 1
− (0)(−senx)] =
− ´ tanxdx = −[−ln(cosx)] = ln(cosx) = 1 u2 = ;
´
dx = x
y p = u1 y1 + u2 y2 y p = ln(cosx)cosx + xsenx y(x) = yh + y p y(x) = c1 cosx + c2 senxi + cosxln(cosx) + xsenx 2
R
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y + y = senx
.
yh = y + y = 0
m2 + 1 = 0 m2 = m1,2 = D
o
n
d
e
:
±√−1
−1 ;
m1,2 =
±i
α=0
β =1
y
yh = eαx (c1 cosβx + c2 senβx) yh = e0x (c1 cosβx + c2 senβx)
2
2
yh = c1 cosx + c2 senx D
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n
i
m
o
s
y1 y2
,
y1 = cosx y1 = ;
−senx
y2 = senx y2 = cosx ;
C
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c
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l
W
r
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k
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a
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o
.
cosx senx = cos2 x + sen2 x = 1 senx cosx
W =
−
0 senx = senx cosx
W 1 =
cosx 0 = senxcosx senx senx
W 2 = A
h
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a
c
a
l
c
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−
u1 u2
,
−sen2x
.
2
− sen1 x = −sen2x ´ u1 = − sen2 xdx = x2 − 14 sen2x u1 =
senxcosx 1
u2 =
senxcosxdx = 12 sen2 x
´
u2 =
= senxcosx
− 14 sen2x)cosx + 12 sen2x(senx) y p = 12 xcosx − 14 cosxsen2x + 12 sen3 x
y p = u1 y1 + u2 y2 = ( x2
y(x) = y p + yh y(x) = c1 cosx + c2 senx + 12 xcosx 3
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y + y = cos2 x
.
a
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− 14 cosxsen2x + 12 sen3x
d
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yh = y + y = 0 l
a
f
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r
m
a
:
yh = c1 cosx + c2 senx D
e
n
i
m
o
s
y1 y2
,
y1 = cosx y1 = ;
−senx
y2 = senx y2 = cosx ;
C
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W
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k
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s
:
2
3
cosx senx = cos2 x + sen2 x = 1 senx cosx
W =
−
0 senx = 2 cos x cosx
W 1 =
cosx senx
W 2 = D
e
n
i
m
o
s
−senxcos2x
0 = cos3 x cos2 x
−
u1 u2
,
−senxcos2x = −senxcos2x
u1 = u1 =
−
1
senxcos2 xdx =
´
u2 =
´
u2 =
− −
cos3 x cos3 x = 3 3
cos3 x = cos3 x 1
cos3 xdx = senx
3
− sen3 x
cos3 x (cosx) + senx 3
y p = u1 y1 + u2 y2 =
cos4 x + sen2 x 3
y p =
−
4
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a
− y = cosh x y − y = 0 m2 − 1 = 0 √ m2 = 1 m1,2 = ± 1 = ±1
a
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y
.
a
d
a
;
yh = c1 ex + c2 e−x D
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n
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o
s
y1 y2
,
y1 = ex y1 = ex ;
y2 = e−x y2 = ;
C
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W
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k
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a
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:
2
4
− sen3 x
cos4 x y(x) = c1 cosx + c2 senx + + sen2 x 3 4
sen3 x 3
−e−x
−
sen4 x 3
(senx)
ex ex
W =
e−x = e−x
−
e−x = e−x
0 coshx
W 1 =
a
l
c
u
l
a
m
o
s
u1
−e−x(coshx) = −e−xcoshx
−
ex ex
W 2 = C
−e−x(ex) − ex(e−x) = −1 − 1 = −2
0 = ex coshx coshx
u2
y
x u1 = −e −coshx = 12 e−x coshx 2 −
1 2
u1 =
´ −x e coshxdx = 18 e−2x (2e2x x − 1) u2 =
ex coshx = 2
−
− 12 excoshx 2x
u2 = y p =
− 12 ´ excoshxdx = − 12 [ x2 + e4
ex [ 18 e 2x (2e2x x
−
2x
− 1)] + (−e−x)(− x − e ) 4
y p = 18 e−x (2e2x x
−x
− 1) + xe4
y(x) = c1 ex + c2 e−x + 18 e−x (2e2x x
4
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y + 3y + 2y =
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d
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−2 m2 = −1
yh = c1 e−2x + c2 e−x
n
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m
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y1 y2
,
.
y1 = e−2x y1 =
−2e−2x y2 = −e−x
;
y2 = e−x C
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k
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:
ex 8
1 1 + ex
(m + 2)(m + 1) = 0
e
;
2
5
8
−x
− 1) + xe4
m2 + 3m + 2 = 0
D
+
yh = y + 3y + 2y = 0
m1 =
]
+
ex 8
e−2x 2e−2x
W =
e−x = (e−2x )( e−x ) (e−x )( 2e−2x ) = e−x e−3x
−
−
−
W 1 = W 2 = C
a
l
c
u
l
a
m
o
u1 =
s
−
−
e−x = e−x
0 1 1+ex
−
−
e−2x 2e−2x
0 1 1+ex
−
−
−e−3x + 2e−3x =
e−x 1 + ex
e−2x = 1 + ex
u1 ,u2
e−x 1 + ex = e 3x −
u1 =
1 e−x = −2x = − x x 3 (e )(1 + e ) e (1 + ex )
− e−2x 1+ e−x
− ´ e−2x 1+ e−xdx = −ex + ln(ex + 1) − 1
e−2x 1 e−2x ex = u2 = 1 + = e−3x (e−3x )(1 + ex ) e−x + 1
´
u2 =
1 dx = x + ln(e−x + 1) − x e +1
y p = (e−2x )[ ex + ln(ex + 1)
− 1] + [x + ln(e−x + 1)](e−x)
−
−e−x + e−2xln(ex + 1) − e−2x + xe−x + e−xln(e−x + 1) y(x) = c1 e−2x + c2 e−x − e−x + e−2x ln(ex + 1) − e−2x + xe−x + e−x ln(e−x + 1) 3y − 6y + 6y = ex secx yh = 3y − 6y + 6y = 0 3m2 − 6m + 6 = 0 a = 3 b = −6 c = 6 y p =
5
.
,
D
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m
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s
,
y1 y2 ,
√36 − 72 √−36 (−6) ± (−6)2 − 4(3)(6) − 6 m1,2 = = ± =1± = 1±i
2(3)
6
6
α=1 β=1 ,
yh = ex (c1 cosx + c2 senx) D
e
n
i
e
n
d
o
y1 y2 ,
2
6
6
y1 = ex cosx y1 = ex cosx ;
− exsenx
y2 = ex senx y2 = ex senx + ex cosx ;
C
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W
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k
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ex cosx ex senx = (ex cosx)(ex senx + ex cosx) ex cosx ex senx ex senx + ex cosx (ex senx)(ex cosx ex senx) = ex (cosxsenx + cos2 x cosxsenx + sen2 x)
W =
−
−
−
−
W = ex (cos2 x + sen2 x) = ex ex senx = (ex senx)(ex secx) = ex senx + ex cosx ex tanx
0 x e secx
W 1 =
−
−
ex cosx ex cosx ex senx
W 2 = C
a
l
c
u
l
cosx 0 = (ex cosx)(ex secx) = ex ( ) = ex e secx cosx x
−
a
m
o
s
−ex( senx )= cosx
u1 u2
,
x
= −tanx − e tanx ex ´ u1 = − tanxdx = −(−lncosx) = lncosx u1 =
ex u2 = x = 1 e
u2 =
´
dx = x
y p = lncosx(ex cosx) + x(ex senx) y(x) = ex (c1 cosx + c2 senx) + ex cosxlncosx + xex senx 2
.
6
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r
− 2y = 0 r
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a
y = xm
.
y = xm y = mxm−1
− 1)mxm−2
y = (m S
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x2 [(m
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.
− 1)mxm−2] − 2(xm) = 0 2
7
x2 [(m
− 1)mxm x−2] − 2(xm) = 0 (m − 1)mxm − 2xm = 0 xm [(m − 1)m − 2] = 0 xm (m2 − m − 2) = 0
a
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m2
−m−2 = 0 (m + 1)(m − 2) = 0 m1 = −1 m2 = 2 ;
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:
y = c1 x−1 + c2 x2 2
.
x2 y + y = 0 S
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y = xm y = xm y = mxm−1 y = (m
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x2 [(m (m
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− 1)mxm−2
− 1)mxm−2] + xm = 0
− 1)mx2xm x−2 + xm = 0 (m2 − m)xm + xm = 0 xm (m2 − m + 1) = 0
m2
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e
:
α=
1 2
β=
1 2
√3
−m+1=0 √ m1,2 = 12 ± 12 3i 1
1
y = c1 x 2 + 2
√ 3i
2
8
1
1
+ c2 x 2 − 2
√ 3i
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xiβ = (elnx )iβ = eiβlnx c
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xiβ = cos(βlnx) + isen(βlnx) x−iβ = cos(βlnx) e
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t
o
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c
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s
− isen(βlnx)
xiβ + x−iβ = cos(βlnx) + isen(βlnx) + cos(βlnx) xiβ
− isen(βlnx) = 2cos(βlnx)
− x−iβ = cos(βlnx) + isen(βlnx) − cos(βlnx) + isen(βlnx) = 2isen(βlnx) s
y = C 1 xα+iβ + C 2 xα−iβ
i
y1 = xα (xiβ + x−iβ ) = 2xα cos(βlnx) y2 = xα (xiβ s
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− x−iβ ) = 2xαisen(βlnx)
y1 = xα cos(βlnx) A
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s
y
= xα sen(βlnx)
y = xα [c1 cos(βlnx) + c2 sen(βlnx)]
√
1
√
y = x 2 [c1 cos( 12 3lnx) + c2 sen( 12 3lnx)] 3
.
x2 y + xy + 4y = 0 S
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:
y = xm y = mxm−1 y = (m S
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x2 [(m
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− 1)mxm−2
− 1)mxm−2] + x(mxm−1) + 4(xm) = 0 xm (m2 − m + m + 4) = 0 xm (m2 + 4) = 0 m2 =
2
9
−4
±√−4 m1,2 = ±2i
m1,2 =
α=0β=2 y = x0 (c1 cos2lnx + c2 sen2lnx) y = c1 cos2lnx + c2 sen2lnx 4
.
x2 y S
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− 3xy − 2y = 0
y = xm y = mxm−1
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y = (m S
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.
x2 [(m
− 1)mxm−2] − 3x(mxm−1) − 2(xm) = 0 xm [(m2 − m) − 3m − 2] = 0 xm (m2 − 4m − 2) = 0 √ m1,2 = 2 ± 6 √ 6
y = c2 x2+ 5
√ 6
+ c1 x2−
.
25x2 y + 25xy + y = 0 S
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y = xm y = mxm−1 y = (m S
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s
.
− 1)mxm−2
25x2 [(m
− 1)mxm−2] + 25x(mxm−1) + xm = 0
xm [25m2
− 25m + 25m + 1] = 0 3
0
25m2 + 1 = 0
m1,2 =
± − 251 = ± 15 i 1 5
α = 0, β =
1 1 y = x0 (c1 cos lnx + c2 sen lnx) 5 5 1 1 y = c1 cos lnx + c2 sen lnx 5 5 6
.
x2 y + 5xy + y = 0 S
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y = xm y = mxm−1 y = (m S
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2 [(m
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s
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− 1)mxm−2
1)mxm−2 ] + 5x(mxm−1 ) + xm = 0 x xm (m2 m + 5m + 1) = 0 m2 + 4m + 1 = 0 m1,2 = 2 √ 3 √ y = c1 x2+ 3 + c2 x2− 3 7
− − ±√
.
− 4y = x4
xy S
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y = xm y = mxm−1 y = (m H
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− 1)mxm−2
e
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− 4xy = x5 yh = x2 y − 4xy = 0 x2 [(m − 1)mxm−2 ] − 4x(mxm−1 ) = 0 x y R
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xm (m2 m 4m) = 0 m(m 5) = 0 m1 = 0 m2 = 5 yh = c1 x0 + c2 x5 yh = c1 + c2 x5
−
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Q(x)y = f (x) D
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− 4 yx = x3
y
x
f (x) = x3 y1 y2 y1 = 1 y1 = 0 y2 = x5 y2 = 5x4 1 x5 W = = 5x4 0 = 5x4 0 5x4 0 x5 W 1 = 3 = 0 x8 = x8 x 5x4 1 0 = x3 W 2 = 0 x3 u1 u2 − x8 u1 = 5x4 ´ = 15 x4 1 5 u1 = 15 x4 dx = 25 x 3 x 1 u2 = 5x´ 4 = 5x u2 = 15 x1 dx = 15 lnx 1 5 y p = 25 x (1) + 15 lnx(x5 ) 5 1 5 y p = 25 x + x5 lnx y(x) = yh + y p 5 1 5 y(x) = c + c2 x5 25 x + x5 lnx i
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− xy + y = 2x
y = xm y = mxm−1
− 1)mxm−2
y = (m R
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yh = x y − xy + y = 0 S
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x [(m − 1)mx − ] − x(mxm−1 ) + xm = 0 m2 m2
m 2
−m−m+1=0 − 2m + 1 = 0
3
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P (x)y +
(m 1)2 m1,2 = 1 yh = c1 x + c2 xlnx
−
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− x1 y + x1 y = x2
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f (x) = x2 y1 = x y2 = xlnx ,
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y1 = 1
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x2 [(m
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m1 = 2 m2 = 1 yh = c1 x2 + c2 x ,
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y − x2 y + x22 y = x2 ex
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− lnx = xlnx −
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∞ c xn n=0 n
y=
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− − − − − − − − − y
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y =
∞ (n−1)nc xn−2 n n=2
∞ (n 1)nc xn−2 x ( ∞ c xn ) = 0 n n=2 n=0 n ∞ (n 1)nc xn−2 ∞ c xn+1 = 0 n n=2 n=0 n
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∞ n(n − 1)c xn−2 − n n=3
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∞ c xn+1 = 0 n=0 n ∞ c xk = 0 k 2c2 ∞ k=1 (k + 2)(k + 1)ck+2 x − k=1 k−1 k 2c2 ∞ k=1 [(k + 2)(k + 1)ck+2 − ck −1 ]x = 0
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∞ c xn+1 = 0 n=0 n
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s
∞ c xn+1 = 0 n=0 n
1)cn xn−2
k=n 2 n = k+2 n = k :
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∞ n(n n=3
2(1)c2 x0 2c2
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2c2 = 0 c2 = 0 ;
c0 3(2)
k = 1 c3 = ,
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1 1 72 ( 180 )c0
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← c5 = 0
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i
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1 12960 c0
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← c8 = 0 r
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y= c0 +c1 x+c2 x2 +c3 x3 +c4 x4 +c5 x5 +c6 x6 +c7 x7 +c8 x8 +c9 x9 +c10 x10 +c11 x11 +..., y= 1 1 1 1 1 c0 +c1 x+0+ 16 c0 x3 + 12 c1 x4 +0+ 180 c0 x6 + 504 c1 x7 +0+ 12960 c0 x9 + 90(504) c1 x10 +0 y = c0 (1 + 16 x3 +
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1 9 12960 x ) + c1 (x
+
+
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1 7 504 x
+
1 10 90(504) x )
2
y + x2 y + xy = 0
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:
n y= ∞ n=0 cn x ∞ y = n=1 cn nxn−1 n−2 y = ∞ n=2 (n − 1)ncn x
− E
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1)ncn x − + x2
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cn xn ] = 0
∞ (n − 1)nc xn−2 + ∞ c nxn+1 + ∞ c xn+1 = 0 n =0 n n=2 n=1 n ∞ c nnx n−2 n+1 n+1 2c2 x0 + 6c3 x ∞ (n 1)nc x + + c0 x1 ∞ =0 − n n=4 n=1 n n=1 cn x
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k+2−2 2c2 x0 + 6c3 x ∞ + ∞ 1)xk−1+1 + k =2 (k + 2 1)(k + 2)ck+2 x k =2 ck−1 (k k−1+1 c0 x 1 ∞ =0 k=2 ck−1 x ∞ 2c2 + 6c3 x + c0 x k=2 (k + 1)(k + 2)ck+2 xk + ck−1 (k 1)xk + ck−1 xk = 0 2c2 + 6c3 x + c0 x ∞ 1) + ck−1 ]xk = 0 k =2 [(k + 1)(k + 2)ck +2 + ck−1 (k (k + 1)(k + 2)ck+2 + ck−1 (k 1) + ck−1
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2c2 = 0 c2 = 0 6c3 + c0 = 0 c3 = 16 c0 ;
−
kck−1 (k + 1)(k + 2) k = 2, 3, 4,... 2c1 1 c4 = 3(4) = 6 c1 3c2 c5 = 4(5) =0 c2 = 0 4c3 2 1 c6 = 5(6) = 15 ( 16 c0 ) = 45 c0 5c4 5 1 5 c7 = 6(7) = 42 ( 6 c1 ) = 136 c1 6c5 6 c8 = 7(8) = 56 (0) = 0 7c6 7 1 7 c9 = 8(9) = 72 ( 45 )c0 = 72(45) c0 8c7 4 5 5 c10 = 9(10) = 45 ( 136 c1 ) = 45(34) c1 9c8 9 c11 = 10(11) = 110 (0) = 0 10c9 5 7 7 c12 = 11(12) = 66 ( 72(45) c0 ) = 66(72)(9) c0 [(k 1)+1] ck 1 (k +1)(k +2)
−
ck+2 = S
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−
y = c0 + c1 x + c2 x2 + c3 x3 + c4 x4 + c5 x5 + c6 x6 + c7 x7 + c8 x8 + c9 x9 + ... 5 5 1 6 7 7 y = c1 [ 16 x4 + 136 x7 + 9(34) x10 ] c0 [ 45 x + 72(45) x9 + 66(72)(9) x12 ] 3
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− y − 2xy + y = 0
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n y= ∞ n=0 cn x ∞ y = n=1 cn nxn−1 y = ∞ 1)ncn xn−2 n=2 (n
− − −− −− E
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∞ (n 1)nc xn−2 − 2x ∞ c nxn−1 + ∞ c xn = 0 n n n=2 ∞ (n 1)ncn xn−2 − 2 ∞ n=1 ∞ c nx=0 n n c nx + = 0 n n=2 n=1 n n=0 n ∞ ∞ ∞ n−2 n 2c2 n=3 (n 1)ncn x − 2 n=1 cnnx + c0 n=1 cnxn = 0
−
k=n 2 k=n ∞ c xk = 0 k +2−2 k 2c2 ∞ (k + 2 1)(k + 2)c x 2 ∞ k+2 k k=1 k =1 ck kx + c0 ∞ c kxk + ∞ c xkk=1 k 2c2 + c0 ∞ (k + 1)(k + 2)c x + = 0 k +2 k =1 k =1 k k=1 k k k k 2c2 + c0 ∞ (k + 1)(k + 2)c x 2c kx + c x = 0 k k k +2 k =1 ∞ k k k 2c2 + c0 k=1 (k + 1)(k + 2)ck+2 x 2ck kx + ck x = 0 H
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2c2 + c0 = 0 c2 = 12 c0 (k + 1)(k + 2)ck+2 xk 2ck kxk + ck xk = 0 [(k + 1)(k + 2)ck+2 2ck k + ck ]xk = 0 (k + 1)(k + 2)ck+2 2ck k + ck (2k + 1)ck ck+2 = (k + 1)(k + 2) k = 1, 2, 3, 4,...
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− −
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−
3c1 2(3)
c3 = 5c2 3(4)
c4 =
c6 =
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c7 =
11c5 6(7)
c0
4
=
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=
c9 =
15c7 8(9)
=
15 11 72 ( 6(40) c1 )
+
5 4 24 x
+
18c9 10(11)
=
=
161 72(240) c1
1 161 55 ( 8(240) )c1
11 7 240 x
1 6 16 x
+
161 161 + 72(240) x9 + 55(8)(240) x11 17(13) 13 8 10 56(16) x + 90(56)(16) x
−
.
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d
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:
−y = 0
y =
∞ c xn n=0 n ∞ c nxn−1
y=
n=1 n
∞ (n − 1)nc xn−2 n n=2 n−2 n−1 − ∞n=0 cnxn = 0 (x2 + 2) ∞ + 3x ∞ n=2 (n − 1)ncn x n=1 cn nx ∞ (n − 1)nc xn−2 +2 ∞ (n− 1)nc xn−2 + ∞ 3c nxn − ∞ c xn = n n n n=2 n=2 n=1 n=0 n y =
x2
11 6(40) c1
17(13) − 9(10)(56)(16) c0
=
(x2 + 2)y + 3xy S
=
13 − 161 c0) = − 56(16) c0
17c8 9(10)
+
7 40 c1
− 245 c0) = − 161 c0
13 56 (
7 5 40 x
=
9 30 (
=
1 3 2x
7 1 20 ( 2 c1 )
=
13c6 7(8)
c11 = y = c1
− 12 c0) = − 245 c0
c8 =
c10 =
5 12 (
=
7c3 4(5)
c5 =
= 12 c1
0
3
7
∞ (n − 1)nc xn + ∞ 2(n − 1)nc xn−2 + ∞ 3c nxn − ∞ c xn = 0 n n n n=2 n=2 n=1 n=0 n ∞ (n − 1)nc xn + ∞ 2(n − 1)nc xn−2 + 3c x ∞ 3c nxn − c + n n n 1 0 n=2 n=2 n=2 ∞ n c x c x =0
− − − − − − − − − − 1
n=2 n
3c1 x + c0 + c1 x ∞ 1)ncn xn + 2(2 1)2c2 x2−2 + 2(3 n=2 (n ∞ c xn = 0 n 1)3c3 x3−2 ∞ 1)ncn xn−2 + ∞ n=4 2(n n=2 3cn nx n=2 n 3c1 x + c0 + c1 x + 4c2 + 12c3 x ∞ =2 (n ∞ 3c nnx n n n=2 H
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k=n
2
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− 1)ncnxn−2 + e
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∞ [(k 1)kc + 2(k +1)(k + 2)c + 3c k − c ]xk = 0 k k+2 k k k=2
4c1 x + c0 + 4c2 + 12c3 x e
d
1)ncn xn + ∞ 2(n ∞ c xn = 0n=4 n=2 n
∞ (k 1)kc xk + ∞ 2(k + 2 − 1)(k + k k=2 ∞ ∞ c kx=2 k k 3c kx =0 k k=2 k=2 k
3c1 x + c0 + c1 x + 4c2 + 12c3 x 2)ck+2 xk+2−2 +
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4c1 + 12c3 = 0 c3 = c31 c0 + 4c2 = 0 c2 = c40 (k 1)kck + 2(k + 1)(k + 2)ck+2 + 3ck k ck 3kck + ck (k 1)kck [ 3k + 1 k 2 k]ck [ 4k + 1 k2 ]ck ck+2 = = = 2(k + 1)(k + 2) 2(k + 1)(k + 2) 2(k + 1)(k + 2) k = 2, 3, 4, 5,... c2 = c40 c3 = c31 −4−2)c2 = 11 ( 1 c ) = 11 c c4 = (−6+1 2(3)(4) 24 4 0 96 0
− −
−
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t
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− −
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− − − − − − −
− − − −
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9)c3 1 1 1 c5 = ( 12+1 = 20 2(4)(5) 40 c3 = 2 ( 3 c1 ) = 6 c1 16)c4 31(11) 31 11 c6 = ( 16+1 = 31 2(5)(6) 60 c4 = 60 ( 96 c0 ) = (60)(96) c0 25)c5 1 11 c7 = ( 20+1 = 8444 c5 = 11 2(6)(7) 21 ( 6 c1 ) = 126 c1 36)c6 11(31) 59 11(31) c8 = ( 24+1 = 112 ( 60(96) c0 ) = 112(60)(96) c0 2(7)(8) 31(11) 6 11(31) 1 3 4 8 y = c0 [ 14 x2 + 11 90 x + 60(96) x + 112(60)(96) x ] + c1 [ 3 x
− − − −
−
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d
a
d
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L
a
p
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a
c
e
+ 16 x5 +
.
−
≤t<1 ≥1 L{f (t)} = ´ 0∞ e−stf (t)dt = − ´ 01 e−st(1) + ´ 1∞ e−st(1) = − e−s |10 + e−s |∞ 1 f (t)
1, 0 1, t
st
st
−
−
3
8
11 7 126 x ]
s(1)
s(0)
s(1)
s(∞)
− e −s − [− e −s ] + e −s − e −s = e s − 1s + e s + 0s = 2es − 1s −
=
−
−
s
−
s
−
−
s
−
2
.
f (t) =
t, 1,
0
≤t<1 t≥1
L[f (t)] = ´ 0∞ e−stf (t) = ´ 01 e−sttdt + ´ 1∞ e−st(1)dt = − e s (t − 1s )|10 + − e s |∞ 1 e e 1 = − s (1 − s ) − [− s (0 − 1s )] + [− e s − e s = − e s + es + s1 − 1s st
st
−
−
s(1)
s(0)
−
s(∞)
−
s
s(0)
−
−
]
s
−
−
2
2
f (t) = te4t
L{te4t } = ´ 0∞ e−stte4tdt = ´ 0∞ te−(s−4)t dt −(s−4)t
− e(s − 4)2 [−s + 3]|∞0
=
e−(s−4)∞ (s 4)2
=
− −
−− −
0
=
(s
−
1
+
4)2
(s
− 4)2
1
= 3
e−(s−4)0 [ ] (s 4)2
(s
− 4)2
.
y + 3y + 2y = 0
y(0) = 1 y (0) = 1
,
A
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f
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n
L[y2 ] + 3L[y] + 2L[y] = 0 [s Y (s) − sy(0) − y (0)] + 3[sY (s) − y(0)] + 2[Y (s)] = 0 s2 Y (s) − sy(0) − y (0) + 3sY (s) − 3y(0) + 2Y (s) = 0 s2 Y (s) − s(1) − 1 + 3sY (s) − 3(1) + 2Y (s) = 0 s2 Y (s) − s − 1 + 3sY (s) − 3 + 2Y (s) = 0 Y (s)(s2 + 3s + 2) − s − 4 S
F
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(4 + s)(s + 1) A= (s + 2)(s + 1) (4 + s)(s + 2) B= (s + 1)(s + 2) 3 Y (s) = s+1 S
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3 y(t) = −1 [ ] s+1 1 y(t) = 3 −1 [ ] s+1 y(t) = 3e−t 2e−2t
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xy = 0
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y =
∞ (n−1)nc xn−2 n n=2
∞ (n 1)nc xn−2 x ( ∞ c xn ) = 0 n n=2 n=0 n ∞ (n 1)nc xn−2 ∞ c xn+1 = 0 n n=2 n=0 n
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∞ c xn+1 = 0 n=0 n ∞ c xk = 0 k 2c2 ∞ k=1 (k + 2)(k + 1)ck+2 x − k=1 k−1 k 2c2 ∞ k=1 [(k + 2)(k + 1)ck+2 − ck −1 ]x = 0
(k + 2)(k + 1)ck+2 ck+2 =
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ck−1 (k + 2)(k + 1) 4
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∞ c xn+1 = 0 n=0 n
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2c2 = 0 c2 = 0 ;
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=
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=
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y= c0 +c1 x+c2 x2 +c3 x3 +c4 x4 +c5 x5 +c6 x6 +c7 x7 +c8 x8 +c9 x9 +c10 x10 +c11 x11 +..., y=
1 1 1 1 1 c0 +c1 x+0+ 16 c0 x3 + 12 c1 x4 +0+ 180 c0 x6 + 504 c1 x7 +0+ 12960 c0 x9 + 90(504) c1 x10 +0
y = c0 (1 + 16 x3 + 2
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∞ (n − 1)nc xn−2 − (x + 1) ∞ c nxn−1 − ∞ c xn = 0 n n=2 n=1 n n=0 n ∞ (n − 1)nc xn−2 − ∞ c nxn − ∞ c nxn−1 − ∞ c xn = 0
n=2
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∞ (k + 2 − 1)(k + 2)c xk ∞ c kxk ∞ c (k + k +2 k =1 k=1 k k=1 k+1 ∞ 1)xk − k=1 ck xk = 0 k c2 − c1 − c0 + ∞ k=1 [(k + 1)(k + 2)ck+2 − kc k − (k + 1)ck+1 − ck ]x = 0
− c1 − c0 +
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c2 = 0 c1 = 0 c0 = 0 (k + 1)(k + 2)ck+2 ck+2 =
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(k + 1)ck+1 + (k + 1)ck (k + 1)(k + 2)
k = 1, 2, 3, ..., k = 1 c3 = =0 3 k = 2 c4 = = 12 c3 = 0 k = 3 c5 = =0 S
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y = x + 2y2 y(0) = 0 y (o) = 1 D
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y = 1 + 4yy y = 4y y + 4yy yiv = 4y y + 4y y + 4y y + 4yy = 12y y + 4yy y v = 12y y + 12y y + 4y y + 4yy iv
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y vi = 12y y + 12y y + 12y y + 12y y iv + 4y y + 4y y iv + 4y yiv + 4yy v = 36y y + 12y y iv + 4y y + 4y yiv + 4y y iv + 4yy v y (0) = 1 y (0) = 4 y iv (0) = 12 y v (0) = 76 yvi (0) = 408 ,
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f (t) = 4t
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´ ∞
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∞
5 1 5
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t (1 5s) 5 − 1 5 0 (1−5s) 5 1 5
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f (t) = et−2
L[f (t)] = L[et−2] 4
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´ ∞ t−2 −st −2 ´ et(1−s) dt e e dt = e 0 t(1 s) −2 )( e( )(1 s) ) − (e−2 )( e(0)(1 s) ) = e−2 [ e 1−s ]|∞ 0 = (e 1−s 1−s e−2 e−2 =
−
=
4
∞
−
−
−1 − s = s − 1
.
f (t) = et cos t
L[f (t)] = L[et cos t] L[f (t)] = F (s) at L[e f (t)] = F (s − a) P
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L[cos t] = s s+1 =⇒ L[et cos t] = (s − 1)s 2 + 1 2
5
.
f (t) = e−t cos t
L[f (t)] = L[e−t cos t] L[f (t)] = F (s) at L[e f (t)] = F (s − a) a = −1 ´ L[cos t] = 0∞ e−st(cos t)dt u = cos t du = − sin t ´ P
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st dv = e−st v = e−st dt = − e s ´ ∞ −st e−st ∞ −
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u = sin t du = cos t st dv = e−st v = e s e−∞ e−0 = [(cos ) (cos 0) ] s s ,
−
∞
1 s2
´ ∞ −st e cos tdt
0
A
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0
´ ∞ −st e cos tdt = ∴ t
e
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1 s
1+ a
d
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=
1 s2 t
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st
st
− e s )|∞0 − ´ 0∞(− e s 1 1 ∞=1 = 2 st sin t|∞ cos t | 0 − 0 s e sest s
−
= 1+
−
−
,
s
− 1s
s +1
s2 l
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c
−
sin t(
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L[f (t)] = F (s) L[eat f (t)] = F (s − a) 4
4
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d
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a=
−1
L[e−t cos t] = (s +s 1)+21+ 1 6
.
−
1 0 1
f (t) =
0
≤
≥
L[f (t)] = ´ 0∞ e−stf (t)dt = ´ 02 e−st(−1)dt + ´ 24 e−st(0)dt + ´ 4∞ e−st(1)dt ´ 2 −st 1 −st 2 1 −2s 1 0 1 −2s − − − − se = s e − 1 e ( 1)dt = [ e ]0 = e 0 s s
´ 4 −st e (0)dt = 0 2 ´ ∞ −st 1 −st ∞ 1 −∞t 1 4s 1 4s − | − e (1)dt = e = e + e = e 4 4 s
L[f (t)] = 1s 7
e−2s
s
−1
s
s
1 1 −2s + e4s = e + e4s s s
−1
.
f (t) =
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0
≥
L´ [f (t)] = ´ 01 e−st´ 3tdt + ´ 1∞ e−st(0)dt 1 e st 3tdt 0
1 0
−
=3 u = t du = 1
te−st dt
,
e−st − st dv = e v=− s e−st 1 ´ 1 e−st ,
−
= 3 t(
s
e−st 1 )dt = t( ) s s 0
)0
| − 0 (−
3 = t( e−st ) 10 s
−
1 −st 1 [e ]0 s
|−
3 3 [ e−s 1s e−s ] = e−s ( s ´ s ∞ 0dt = 0 1 3 1 [f (t)] = e−s ( 1 + s s
−
8
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− −
c
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t
r
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r
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− =
1 [( e−s ) s
− 0] − 1s [(e−s) − 1]
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f (t)
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F (s) ,
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f (t) = −1 F (s)
L {
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L−1[ s1 ] L−−11[ s1n! ] = f (t) = tn L [s ] = t 2
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f (t)
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f (t) = t sin3t f (t) = t sin3t f (0) = 0 f (t) = sin 3t + 3t cos3t f (0) = 0 f (t) = 3cos 3t + 3cos 3t 9t sin3t f (0) = 6 [f (t)] = s2 F (s) f (0) f (0) [3cos3t + 3cos 3t 9t sin3t] = s2 [t sin3t] 0 6 [cos 3t] 9 [t sin3t] = s2 [t sin3t] 6 [cos 3t] = s2 [t sin3t] + 9 [t sin3t] 6 [cos 3t] = (s2 + 9) [t sin3t] s 6[ 2 ] = (s2 + 9) [t sin3t] s +9 6s [t sin3t] = 2 (s + 9)2 s
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L L L L L
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− L L
− −
L L
− − L L
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L
L 1
0
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f (t) = t cosh t s
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:
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L − − L L L L L L L −L − L − − − L − − − L − L − ,
− − − −
4
6
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p
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1
1
.
f (t) = t2 cos3t S
e
a
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:
f (t) = t2 cos3t f (0) = 0 ,
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A
p
l
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c
a
n
d
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e
l
t
e
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r
e
m
a
,
L[f (t)] = s2F (s) − f (0) − f (0) L[2cos3t − 12t sin3t − 9t2 cos3t] = s2L[t2 cos3t] − 0 − 0 2L[cos3t] − 12L[t sin3t] − 9L[t2 cos3t] = s2 L[t2 cos3t] 2L[cos3t] − 12L[t sin3t] = s2 L[t2 cos3t] + 9 L[t2 cos3t] s s2 + 9
2
t
r
a
n
= (s2 + 9)L[t2 cos3t] ⇐ − 12 (s2 6s 2 + 9)
s
f
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a
d
a
d
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t sin3t
e
6s + 9)2
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2s(s2 + 9) 36s = (s2 + 9) [t2 cos3t] (s2 + 9)2 2s3 + 18s 36s = (s2 + 9) [t2 cos3t] (s2 + 9)2 2s3 18s = [t2 cos3t] (s2 + 9)2 (s2 + 9) 2s3 18s [t2 cos3t] = 2 (s + 9)3
− −
L L
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L
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t
e
e
l
t
e
o
r
e
m
a
d
e
l
a
t
r
a
F (s) 1
2
.
1
F (s) =
L−1 S
a
b
e
m
1 s(s o
s
− 4) L−1 s −1 a q
u
e
= eat
,
e
n
s(s
t
4
o
7
n
− 4) c
e
s
:
n
s
f
o
r
m
a
d
a
d
e
l
a
i
n
t
e
g
r
a
l
d
a
d
a
L−1
1
=
´ t 0
s(s 4) 4t e 1 1 = = (e4t 4 4 4
−
3
at
|t0 = ea − a1
o
n
d
e
:
a =4
1 + 3)
.
F (s) =
L L L
s2 (s
1 = s2 (s + 3) 1 −1 = eat (s + a) ´ t 1 −1 = 0 e3τ dτ = s(s + 3) ´ t 1 3t 1 −1 = e 0 s2 (s + 3) 3 1 f (t) = (e3t 3t 1) 9
L−1 S
d
− 1)
−
1
eaτ a
eaτ dτ =
i
,
d
o
n
d
e
a=3 ,
e
n
t
o
n
c
e
s
:
1 3τ t 1 1 e 0 = e3t 3 3 3 1 1 3τ 1 dτ = e τ 3 9 3
|
−
−
−
t
1 3t e 9
= 0
− 13 t − 19
− −
1
4
3
.
F (s) =
L−1 C
o
n
o
L−1 L−1
1 t 3
− L − − c
3
s2 (s2 i
e
L
a
p
e
l
1
a
s
c
5
o
a
−1
s2
3
=
9)
3
s2 (s2
o
e
d
l
v
e
r
l
a
s
s
i
g
u
a2
´ t
= sinh at
,
d
o
n
d
e
a=3
,
e
n
t
o
n
c
e
s
:
1 1 1 cosh 3τ t0 = cosh3t 3 3 3 1 1 1 1 cosh 3τ dτ = sinh 3τ τ 3 3 9 3
sinh3τ dτ =
0
´ t
=
− 9)
1 (sinh3t 9
− 9)
=
9)
s(s2
f (t) =
R
n
s2 (s2
0
|
−
−
t
−
1 sinh 3t 9
=
0
− 3t) i
e
n
t
e
s
e
c
u
a
c
i
o
n
e
s
d
i
f
e
r
e
n
c
i
a
l
e
s
,
.
y + y = 0 , y(0) = 1
L{y + y} = L{0} 4
8
u
s
a
n
d
o
l
a
t
r
a
n
s
f
o
r
m
a
d
a
d
e
−
P
o
r
e
l
t
e
o
r
e
m
a
d
e
t
r
a
n
s
f
o
r
m
a
d
a
s
d
e
d
e
r
i
v
a
d
a
s
s2 Y (s)
− sy(0) − y(0) + Y (s) = 0 s2 Y (s) − s(1) − 0 + Y (s) = 0 Y (s)(s2 + 1) − 1 = 0 S
u
s
t
i
t
u
y
e
Y (s) = A
p
l
i
c
a
o
r
d
o
e
l
v
a
l
o
r
i
n
i
c
i
a
l
1 +1
s2
m
o
s
l
a
t
r
a
n
s
f
o
r
m
a
d
a
i
n
v
e
r
s
a
a
Y (s)
1 = e−t 2 s +1
L−1 p
n
l
i
n
e
a
l
i
d
a
d
y = e−x
1
6
.
y + 4y = 2 , y(0) = 0 , y (0) = 0
L [y + 4y] = L [2] 2 s2 Y (s) − sy(0) − y (0) + 4Y (s) = s S
u
s
t
i
t
u
y
e
n
d
o
e
l
v
a
l
o
r
i
n
i
c
i
a
l
2 s2 Y (s) 0 0 + 4Y (s) = s 2 Y (s)(s2 + 4) = s 2 Y (s) = s(s2 + 4)
− −
s
e
a
p
l
i
c
a
l
a
t
r
a
n
s
f
o
r
m
a
d
a
i
n
v
e
r
s
a
2 −1 = 2 s(s + 4) ω −1 = sin ωt ω =2 2 s + ω2 ´ t 2 1 −1 = sin2τ dτ = cos2τ t0 = 0 2 s(s + 4) 2 1 1 f (t) = cos2t 2 2
L S
L
i
L
d
o
n
d
e
,
e
n
t
−
o
n
c
e
s
− 12 cos2t + 12
|
−
p
o
r
l
i
n
1 y= 2
1
7
e
a
l
i
d
a
d
1 cos2x 2
−
.
y + 16y = 4 ,
y(0) = 1 , y (0) = 0
A
p
l
i
c
a
m
o
s
t
r
a
n
s
f
o
r
m
a
d
a
d
e
L
a
p
l
a
c
e
a
L[y + 16y] = L[4] 4
9
l
a
e
c
u
a
c
i
ó
n
4 s
s2 Y (s) + sy(0) + y (0) + 16Y (s) =
S
u
s
t
i
t
u
i
m
o
s
l
o
s
v
a
l
o
r
e
s
i
n
i
c
i
a
l
e
s
,
y
d
e
s
p
e
j
a
m
o
s
Y (s)
4 Y (s)(s2 + 16) + s + 0 = s 4 4−s2 s 4 s2 s Y (s) = 2 = 2 s = (s + 16) (s + 16) s(s2 + 16)
−
A
p
l
i
c
a
m
o
s
t
r
−
a
n
s
f
o
r
m
a
d
a
i
n
v
e
r
s
a
2 4 s −1 4 s −1 −1 = = s(s2 + 16) s(s2 + 16) (s2 + 16) s −1 = cos ωt ω=4 s2 + ω 2 s −1 = cos 4t 2 (s + 16) 4 −1 2 s(s + 16) ω −1 = sin ωt ω =4 2 s + ω2 t ´ t 4 1 1 −1 = 0 sin4τ dτ = cos4τ = cos4t 2 s(s + 16) 4 4 0 1 1 cos4t + 4 4 4 s2 1 1 3 1 −1 = cos4t + cos4t = cos4t + 2 s(s + 16) 4 4 4 4
L L L E
s
−
n
t
o
n
c
e
L
L
p
o
r
l
i
s
p
a
r
a
e
a
l
i
d
d
d
o
n
d
e
,
d
o
n
d
e
−
a
−L
−
n
L
L
,
L
i
−
−
−
−
1 4
=
−
3 1 y = cos4x + 4 4 1
8
.
y A
p
l
i
c
a
m
o
s
l
a
t
r
a
− 2y + 5y = 0 , n
s
f
o
r
m
a
d
a
d
e
L
a
y(0) = 2 , y (0) = 4;
p
l
a
c
e
a
l
a
e
c
u
a
c
i
ó
n
L [y − 2y + 5y] = L [0] L[y](s) = Y (s) L[y](s) = sY (s) − y(0) = sY (s) − 2 2 2 L[y](s) = s Y (s) − sy(0) − y(0) = s Y (s) − 2s − 4 s2 Y (s) − 2s − 4 − 2 [sY (s) − 2] + 5Y (s) = 0 s2 Y (s) − 2s − 4 − 2sY (s) − 4 + 5Y (s) = 0 Y (s) 2− − − Y (s)(s 2s + 5) 2s 8 = 0 Y (s)(s2 − 2s + 5) = 2s + 8 2s + 8 Y (s) = 2 (s − 2s + 5) U
s
A
l
F
a
s
t
o
m
s
s
t
h
u
o
u
c
A
e
a
o
s
a
r
i
i
c
e
t
m
o
r
u
z
a
r
f
t
o
r
s
i
a
m
r
e
m
a
o
l
o
u
c
s
l
s
t
s
u
s
a
a
n
d
s
y
l
u
a
e
d
m
e
o
d
e
e
s
x
s
s
a
r
e
o
i
f
o
a
l
r
s
e
m
e
o
n
o
o
n
m
r
l
s
s
a
t
r
n
e
j
a
r
r
p
p
l
a
t
s
f
s
d
e
a
n
s
l
d
a
e
e
c
L
u
a
a
p
c
i
l
ó
a
c
e
n
d
n
o
e
s
d
d
a
e
r
i
v
a
d
a
s
o
n
a
r
m
f
r
a
a
d
c
c
a
i
i
o
n
n
v
e
5
e
s
0
r
p
s
a
a
d
r
c
e
i
a
l
l
e
a
f
s
u
n
c
i
ó
n
r
a
c
i
o
n
a
l
Y (s) ,
p
a
r
a
s2 2s+5 (s 1)2 + 22 2s + 8 A(s Y (s) = = 2 2 (s 1) + 2 (s 2s + 8 = A(s 1) + 2B s = 1, 1 C
o
m
o
−
−
e
s
i
r
r
e
d
u
c
i
b
l
e
,
e
s
c
r
i
b
i
m
o
s
e
s
t
e
f
a
c
t
o
r
e
n
l
− − 2(−1) + 8 = A(−2) + 2B 6 = −2A + 2B 2(1) + 8 = A(0) + 2B B = 5 6 = −2A + 10 A = 2 2(s − 1) + 2(5) 2s − 2 + 10 2s + 8 Y (s) = = = (s − 1)2 + 4 (s − 1)2 + 4 (s − 1)2 + 4 a
e
c
e
n
e
m
o
l
p
s
r
i
f
o
r
m
a
(s α)2 +β 2
−
− 1) + 2B − 1)2 + 4
−
H
a
m
e
r
c
a
s
o
o
b
t
e
n
e
m
o
s
,
,
,
S
u
s
a
p
t
l
i
i
t
c
u
a
i
m
m
o
o
s
s
e
t
r
s
a
t
o
n
s
s
f
v
o
a
r
l
m
o
a
r
d
e
s
e
a
i
n
n
e
v
e
l
r
d
s
e
a
d
s 2 −1 + 4 −1 (s 1)2 + 4 (s t t f (t) = 2e cos2t + 4e sin2t
L
1
9
−
L
s
a
r
e
r
L
o
a
l
p
o
d
l
a
e
c
e
2 1)2 + 4
−
f
r
a
c
c
i
o
n
e
s
p
a
A
p
l
i
c
a
m
o
s
l
A
p
s
f
l
o
i
r
L−1
c
a
m
m
a
o
d
s
a
a
t
f (t) = L−1 n
a
l
e
s
.
.
s2 + 1
a
i
−1
s2 W (s) − sw(0) − w(0) + W (s) = L[t2 ] + L[2] 2 2 W (s)(s2 + 1) − s + 1 = 3 + s s 2 2 + 2s W (s)(s2 + 1) = +s−1 s3 2(1 + s2 ) + s4 − s3 3 2 + 2s2 + s4 − s3 s W (s) = =
r
c
w + w = t2 + 2 , w(0) = 1 , w (0) =
t
r
e
s
n
s
f
o
r
m
a
v
e
o
e
r
r
e
s
m
a
a
s
.
a
i
n
v
e
r
+ −1
e
l
a
t
r
a
n
s
s
a
L
−L
2 s −1 + (s2 + 1) s(s2 + 1) f
o
r
m
a
d
a
d
e
l
a
i
n
t
e
g
r
a
l
p
a
r
1 (s2 + 1)
−1
a
l
a
s
d
o
s
p
´ t
2sin τ dτ = 2 cos τ t0 =
−2cos t + 2 =− − −2(sin τ − τ )|t0 = −2(sin t − t) = 2t − 2sin t ´ t 1 = 2 0 (τ − sin τ )dτ = 2( τ 2 + cos τ )|t0 = t2 + 2 cos t − 1 2 ´ t 2 1 1 = 0 (τ + 2 cos τ − 1)dτ = ( τ 3 + 2 sin τ − τ )|t0 = t3 + 2 sin t − t 3 3 ´ 2 t L−1 s(s2 + 1) = 0 2sin τ dτ = −2cos τ |t0 = −2cos t + 2 L−1 (s2 s+ 1) = cos t s3 (s2
=
d
d
L
s3 (s2 + 1) t
n
a
2
l
i
2
r
s3 (s2 + 1)
0
|
+ 1) 2 0 [cos τ 1] dτ =
´ t
5
1
r
i
m
e
r
a
s
