Beginners Guide to TikTok for Search - Rachel Pearson - We are Tilt __ Bright...
Apps.pdf
1. اﻟﺠﺰﺋﯿﺔ اﻟﻜﺴﻮر
Partial Fractions Decomposition
3
2
1
3
2
1
3 2 3
2 2
.
.
4
1
4
4 1 1
1
1 1 1
4 2 2 4 0
0
2
s
Coeff
s
Coeff
s
s
F s
s
s A B C
s A s B s C
s
s s s
s As As A Bs B C As B A s A B C
A
B A
0
.
4 0 4
Coeff
s
B
A
3 2 3
1 1 1 2
2 3 2 3
4 4 4
1 1 1
0 4
4 4 4 4
4 2
1 1
t t
s
s s s
B C C
F s e e t t
s s
s s
2
2
1
0
2
2 2
2
1 4 2
2
.
.
.
1 4
1 4 4
1 4
4 1 1
4
0
1 1 4
1 , ,
5 5 5
4 0
s s
Expand
Coeff
s
Coeff
s
Coeff
s
s
F s
s s
s A Bs C
F s
s s s
s s
s A s Bs s C s
s A B s B C s A C
A B
B C A B C
A C
1
2 2
1 1 1 4 1 1 1 2
cos 2 sin 2
5 1 5 4 5 4 5 5 5
t
s
F s e t t
s s s
2.
1
2
3
2
2
1 3 2
2
3
6
3 9
2 5
3 3
6 3 2 3 2
3 6
3 5
3 9 1 6 1 9 6
6 5 3 5 2 5 5
s
s
t t
s
F s
s s
s
A
s
s s A B
s s s s s s
s
B
s
s
F s e e
s s s s
2 2
2 2
2 2 2 2 2
2
1
2 2 2 2
2
1 4 2 2
2 2
&
3 2
1
1 4
1 1 1
1 4 4 5
4
1 4 1
1 1 1
1 5 4 4
1 4 1
1
1 1 4 4 1 1
5
1
2 4
5
s
s s Expand
Simplify
F s
s s
A B Cs D
B
s s s s
s s s
A Cs D
s s s
s s s
A s s s Cs s D s
A C s A D C s
3
2
1
1
.
.
.
2 2 2
5
2 4 1
4
0
1 2 2 3
2 0
5 25 25 25
4 2 0
2 1
2 3 1 2 1 2 3
25 5 cos 2 sin 2
1 25 4 25 4 25 5 25 50
1
, ,
Coeff
s
Coeff
s
Coeff
s
t t
A C D s A D
A C
A D C A C D
A C D
s
e te t t
s s s
s
3. ﻻﺑﻼس ل ّﻣﺤﻮ ﺗﻄﺒﯿﻘﺎت
Applications of Laplace Transform
Use Laplace Transform to solve following ODEs:
3
2
2
2
3
2
2
2 2
2 2 2 2 2 2 2
2
2 3 0 2 0 4
1 3
4 2 2 4 3
3
3 3
2 1 4 10 1 4 10
3 3
3 1 1 10 3 4 6
4
1
1 3 1 1 1 3 1
3
1
t
s s
Expand
y t y t y t te y y
s Y s sY Y
s s
Y s s s Y s s
s s
s
Y s
s
s s s s s s s
s s
, ,
1
2
3
2 2 2
1 3
2
2 2
2
3
3 3 2 2
0 0 2
2
3 2 2 2
3
3
1
3 1
3
3 1 3 1
3
1
2
2
3 3 3 1
2
4
1
0
8
1 1
3 2
2 4
t
t t
t
t t
t
t t t
t
te
s
te te
s s
te
s
t e
t e
te te e t e d e t e d
e
e
e t e t e
2 3 2 2
0
1 3 2 2
1 1 1 1 1
3
4 4 4 4 4
1 1 1 1
3 4 6
4 4 4 4
t
t t t
t t t t t
e e te e t
y t Y s e te e t e te
4.
1
2
2
2
2
3 2 2 3 2 2
3 2
2
2
4 4 , 0 0 , 0 1
1
1 0 4 1 4
2
1 1
4 4 4 2 4
2 2
1 4 1 2 2 4
2 2 2 2 2 2
1 1 2
2
2 2
2
t
Expand
t
y t y t y t e y y
s Y s sY Y
s
Y s s s Y s s
s s
s s
Y s
s s s s s s
Y s
s
s s
t
y t e
1 2t
Use Laplace Transform to solve:
1
2
0
3 3 2
2
Unify
2 3 Denominators 2 3
2 2
3 2 5
3 5
2 4
sin
2 2 1
sin
1
1 2 2
1
1 1
2 1 1
2
1 1
2
1 1
2
2 4!
t
y t t y t d
Y y t t Y
s s s
s
Y Y
s s s s
s s
Y s
s s s
Y s
s s
y t t t
2 4
1
12
t t
5.
0
2 2 2
2
Unify
2 2 Denominators 2 2
2
2 3 2 3
cos 0 4
1 1
4 cos
1
1 1 1
1 4 4
1 1
1 1 1 1 1
4 4
t
y t t y t d y
s
sY y t t Y
s s s
s
sY sY
s s s s
s
Y s
s s s s s
Y s
,
1
5 3
4 2
1 5 4
5
4
4! 2
s s s
t t
y t
0
0
2 2
Unify
2 2
Denominators
2
2
2
1 1 1
2 2
1
1 1
2 1 1 1
1
1 1
1 1
1 1 1
1 1 1
1
t
t t
t
t t
t
Partial
y t te e y e d
y t te y e d
Y y t e Y
s
s s
s
Y Y
s s
s s
s
Y s
s s s
s
1
1 1
2 2
1 1
1 1
2 2
Fractions
t t
Y s
s s
y t e e
6.
1
0
2
3
Unify
2 Denominators 2
2
3 3
2
cosh 0 0 1
1 cosh 0
1 0
1
1 1
1 1
1 1 1
1
1
2
t
y t y t d y
sY y t t
s
sY Y
s
s s
Y s Y
s s
s
Y s
s s s
y t t
,
7.
2
2 2
2 2
2
2 2 2
1
2
2
2
1
, 0 1 , 0 0 , 1 , 0
2
1
1 1
0 1
1 1
1 1
1 1 1
1 1 1
1
sin
1
1 1 1
1 1
1
s
s
s
y t y t f t y y f t t
f t u t
s Y s Y e
s s
Y s e
s s
Y s e
s s s s s
t
s
s s
s s
s
0
0
1 1
2
2 2
2 2
2
2
1 sin
1
1 sin sin cos 1 cos
1 1
1 cos
2
1 1
1 cos 1 cos sin
2
t t
s
t t
t
t d t
e u t u t t
s s s s
y t t u t t t
8.
2
2 2
2 2 2
2
10 24 120 , 0 0 , 0 5
120
5 0 10 5 24
120 120
10 24 5 50 6 4 5 50
120 5 50
6 4 6 4
s
Expand s s
s
Partial
Fractions
y t y t y t u t y y
s Y s sY Y e
s
Y s s s e Y s s e s
s s
s
Y e
s s s s s
1
2
6 2 4 2 6 4
2
10 15 5 8 15
6 4 6 4
10 15 5 8 15
s
t t t t
Y e
s s s s s
y t u t e e e e