Chapter : Differential Equations C DES ) Algebraic : e. s . ' . 3. x - - 4 ⇒ }3x=tz4 ⇒ 11=413 a number malt = e.g. : Cos Of sift = 42 ⇒ ① ? fan ? Are we correct ? DEF : fan - - x 't Xtc e.g . FIH = 3×71 ⇒ ⇒ ftp.3xtl in derivative fix , = Sftxldx = S 3 x' tidy = x 't Xtc In general solos are NOT UNIQUE !
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Chapter : Differential Equations C DES )
Algebraic:
e.s .
'
.
3.x-
- 4 ⇒}3x=tz4⇒ 11=413
a numbermalt =
e.g. :Cos Of sift = 42 ⇒ ① ?
fan ? Are we correct ?
DEF : fan -
- x 't Xtc
e.g .
FIH = 3×71 ⇒⇒ ftp.3xtl
in
derivative fix , = Sftxldx= S 3 x' tidy = x 't Xtc
In general solos are NOT UNIQUE !
we have be solving DE s all tem-
!
EI ! daff = cos x t X
⇒ ya , = Sd#d× = Scosxtxdx = sin xtxfc
E : dat -
-
sink )
⇒ ya = S¥×d× = f sin oh,
NO CLOSED
FORM Soon !
Ex'
. Suppose SCH is the position of a particleat time too
.
If UCH = t 't I and the
particle is initially at 4,
or* a
-
HiaFind SHI
.EYESant .
UH -
- It # D= s'
Ct )rate of
velocity change of position ?
SHI -
- Ss' Aloft = Such It = St 't Idf -
- ft"tt+⑨
4=567--4105't Otc⇒
C=4_⇒ ISHI= It'
't t t 4
-
I
Setting up a DE- .
Exe.
Compound Interest C continues ) rt
Ao -initial amount
⇒ Aft ) = Aoe
r -interest rate
Note daft = Ao . rent = r Ao e"
= r Act ,
or dfrso 4¥ is directly
proportional to A.
checkings-at.ms
ex
:I=f¥Pa) Is Pct ) = 50 t ett a solution ?
do=ot¥et=fse" Yes it is sdn !t= ¥60 - ie"5t ) - to
= lot te" 't -10=fr
b) Is PCH = 50 t Ce" 't
a so In ?
doff --
Ot c. Is e' "- t
= Is cells 't
FP - to = Yottce' "
- t-10
= Ice's - t
For anyvalue of C
,PCH is a soda.
Separation of Variables-
Exel.
Find ACH where doff = r A
manipulate to get A onthe left and t on the risk
.
dat = r A ⇒ dad = rdt
W T.ly involves t
only invokes
A
⇒ In IAI = S dad = Srdt = rt t C
what =
CrttoehD=B
, B > 0e e
Go
rttc lueth B , any box
1A , = e=Eeat et7.e
"
rt.
IAF Ce1×1=2
Acts for.tt
11=12
or
ACH -.
Cert C ? Ao -- Aco ) -
- C. E' I c. etc
or Acth Aoe't
Important DES ( InteEst ) Einstein 's Field Equations- -