Algebraic }3x=tz4 - University of Massachusetts...

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- -

Newton's 2nd Law-

Fluid Mechanics

Quantum Mechanics-

-