Section 1.2
_
first
order
Linear
Must
Be
function
Of
↓
pa
\
y
a ,( x
)
(
) ☆+
出 +a
0 ( 0 ) / = FD
tí
0+ 1
cos ( 6)
261 =
Ex
udr + (v + uv
-
np4)du
0
=
difirputials
4
v
function
a
岳
u
u
u
a
of
出 ( itu)
+
(v
+ uv-
+
ueu
v =
function
Not
‰
v
up4)d
=
0
Limpar
C is
Uprify
is
a
the
parampter
one
solution
y
VtuV - u : o
+
u
a
and
G
of
=
LHS :
出
RUS :
2(2024
arbitrary const
a
family
y
=
cp",
-
x(XX
24
C 0 % (2) = 2 C 00
=
LHS = RHS
e) symBoLices
tho n
se
Problem
다
2
=
Y cosy
y:
,
2) 12) (l-Sinx(
LHS :
*
-
(
RUS :
/l-sinx)
1 20sX)
-
)k
; ( cos ; )
l - s' n
(11-sinx( ) cosy
*
(
)k
; ( cos ; )
I - sin
LUS
RUS
=
□
=
"
fa ( i sinoj
"
-
、
PXcluds
all
X
」
Sinx = 1
,
Y =
( + 2πk
Where
R
is
Domain
A
integer
a
=
domain
(ん
value
R &* + 2m:nez]
as
、
*
a
s )
_
*
solution
Verify
-
2x2y + y
1
=
is
成
⑥(
武(
2
-
∴∴~
(
)
,~
of
に
哉
装
(6
26
=
+
+
5
~
-
歳
+
+2成
28 % +
=
\
24 %
-
)
)
0
0
0
=
:
0
=
~% +~= 1
+ 2
(
歳
)
r∞
-; )
~
2
-
^÷+ 0
㎡
」
solution
=
-
20 +
-
implicit
a
(~ ;) di 0
2 / 5 dI +
0
) =∞
)
다
☆
26
0= 8
2 < cos !
=
6
-
%。
↓
S
2
Lus :
% 15。 + 崎外 * 州盛
n
τ。
が
( )- i以 ]
”
]
忙
。
IPCOSO +
!
。
I」 o、
!
。
2XLos X
RUS :
2x cos &
LHS
RHS
=
□
Ex
숨
=
p
미미
.
=
LH 5 :
0
%
哉
o
い'
: Liec
Rus
“
r
:Gt)l- =
LHs
=
RHS
□
Domain
Casel
case 2
of
pas
ther
Cas
solution
each
CO
-
exclude
け CO
「
for
(
-
for
120
domain
山
1-
0
,
亡
∞
,
domain
∞
of
is
)
t = in)
then
1( % ))
solutions
:
Be
the
ctsx
0
=
the
>O
∞
all
cpta 1
一
一
一
Ex
must
excluded
the
or
solution
kniy) 0)
,
where
ตั
π
↑=㎡+ 25 3
-
is
there
Y
"l"such
a
=
is
C
that
solution
a
is
y:c
y =
solution
a
c + 22 -3
(2+312- 1
0=
C
-
3
-
c =
1
Ex
Determine
is
Y"-y' - 201
solution
a
y
that
So
a
=
ph
,
-x(x) *
0
=
7 "= might = mpms
ว" - Apr -200 =
(Mem -2o) &
-M - 20 = -
(m s((r 4)
for
0
=
+
-
RUs = LHS
Mas
m
=
9
-
π
find m
So
for
its
socution
a
Lauchy
puler's
~
」 y
T'
=
2/ 1
ー
y = /1
、
T
MJB
0
=
"
=
m
(m 1 ) An
lele 1(ye -2) 2x(exn 1)
-
-
-
m(e 1)x2
-
-
21xe
( - sm
^
(
m
-
-
z
.
OCXCA
OCXCA
↑
to
parations
100
-
occo
s
)(
-
m+z
LuS
if
.0
=
M= S
-
-
10x =
0
10x" = 0
)!
m
0
=
) i ^= 0
RUS
or
m
-
2
8n0(X10
Picar '
$
Pfistone
arkil
por
& UP
ㅣ
florkd
os
initial
R
value
f is
Cont
in
R
p is cont
in
R
1)
Hostd
-
problem
then
therp exists
socution
a
E」
Picards
Apply
= an
thi
λ (s ) 의
분 회
도
=
(:>
,
"
9 ×
-
%까
×
品
cont
㉛
picards
thrm
the
Prob
=
S
8.
)
6
치
Has
a
solution
E,
Picards
Apply
- ar
thi
114
such
1 .: }
3
=
Rpetendal
a
must
contain
whprp
'
Grater
then
Point
a
Gord
#
is
that
point
So the
thre
3
.
is
the function
undefined
at
Cannot BP
uSPd
E,
( U R ) J: 1
. 114 ) =
-
0
in
'
BP
qn
2
に
2
·
=2
·
(40 ,% )
㎡
·
the
(40 ,% )
Regions
1
) 472
)
2
-
2 < /E 2
3) Y7
(40 ,% )
-
2
arp
Regions
Should Be
that
j
What
Picards
applipd
10
,
%
)
found
So
thin
can
ー
*
=
f(x, y)
one
of
of
the
curup
-\
Given
of
can
famisy
solutions
List
a
tangents
you
Visualize
Pht
function
parameter
≈
PVplopp
Autonomous EQ
Phase
PQUiLiBriUm
Point
diagrams
S decipasing
: 9%
^
n
d!
π incrpasing
3
Staßle
PquiliBrium
Point
(attractor)
Δ
-dicreasing unstable Pquilibrium Point
(Repplipn + )
方
一
Section 2.1
Toperabele
pavations
Ex
哉
+ 20 ~= 0
出
こー2 ル
dJ =
21 rdy
-
- d%
2 ) d )
-
ftdi fids
- -
TI
}
J
- ℃+ c
:
≥
=
2
+ q
c
」 =
Ex
—
+ 26-
—
-
6
0
…∴
-3 + / s
6
ル 21
)
、 -110 +
+
)
16 0 -
+
- 3
0
"
出 (
- 10 +
(
(
3)
d
1 -"
:
( 6 3 ) (0
-
茂
5
崎
:
)
*…
“
無い 。
成
無d f
=
。
歳示北 + 前
=%
+ 22 mt い
22 n \ t
11
5 hn 1 t 3 引 = 1
-
-
川器|=
1|Fpo
lal
if
we
1
allow
E
y: 1
to
function
ChPCR
:
LUs
=
Y
山
山
E
=
D
—
O
=
-
'
pe
D is
Where
-
arbitrary
positive
- ” ニ EE
,
'
"
o
-
then
s
)
s
family
if
y= 1
is
キO
Eto
Have
we
the
a
0 + 2 x- 6 0
RUS =
I
is
does
Not
a
- 3 ^+
X+ 2
-
X
-
2
…
!
-
3 + 103
solution
Have
added
solution
—
…∴
a
Const
Ee
:
'
p
-
に ± D
-
ite
' ee -
器
(t )
1+ E
-
- tee
t
-
1= Dpo
=± a
が
= 4 gho - 引 + c
있지 ] 에J=
日
小
日
小
Section 2.2
so
Restriction
the
Section 2.3
=ear
parations
normalized
form
A
= Pl × ) )
=1
24 =
p
Ex
*
+
잃 =θ
Ff (
y
solve
-
러
) do
Jo Jayd
dx
=
o-
2
y
"
'- a
=
for
-
/
fiodi
Road BLOCK
Multiply
By p24
all
℃ 背 + 20℃= 0
”
造 ( β = pus
)
)
-
foud
.
-
pt yu puirc
た」
品
p is integrating
factor
โอน + PLAY = Q1
I
{ pd.
0
=
+I
SIl
=
π
+
1
24=
3
出号 に
f
Ed-
-
05
/I
yx
=
'
pounlsl
=
SI ℃
fx
=
-
- x d/
' (
+
=☆+ 臨
lop
=
.