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Integration & First Order ODEs: Slope Fields & Existence, Lecture notes of Mathematics

University of Oklahoma (OU)Mathematics

A review on integrals and the concept of anti-derivatives. It also introduces the method of solving first order ordinary differential equations (ODEs) using slope fields and discusses the existence and uniqueness theorem. Examples are given to illustrate the concepts.

Typology: Lecture notes

2020/2021

Uploaded on 02/09/2022

spencer-smith
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bg1
II1，2Aug 24
Review on Integrals 1.1
aSlope Fields
Existence and Uniqueness
Review on Integrals
If you have taken Calculus you have solved differential
equations
oAnti_derivatives
Afunction Fcx is called an anti_derivative of f
if Fxfcx
We often use indefinite integrals 九deaoteanti
deniatexifcosxdx
s.in xtc
Examples int by parts asub
asolve 毙tet
get ftet It tC
pf3
pf4
pf5

Partial preview of the text

Download Integration & First Order ODEs: Slope Fields & Existence and more Lecture notes Mathematics in PDF only on Docsity!

I I 1 ，^2 Aug 24

Review on^ Integrals 1.

a Slope Fields

Existence and Uniqueness

Review on (^) Integrals If you have^ taken^ Calculus^ you have^ solved^ differential

equations

o (^) Anti_derivatives

A function Fcx is called an anti_derivative of f

if F x fcx

We often^ use^ indefinite^ integrals 九^ deaoteanti

deniatexifcosxdxs.in^ xtc

Examples int^ by parts^ a^ sub a (^) solve

毙 tet

get

ftet

It t^ C

Let u^ t^

2

Then du 2 tdt

tdtzfdyo

Hence ylt

f

⽀ ědutc

⼆

ett⼆^ 点 ceticyltl is called a

gura_lsototheequatnon.gl

H ⼆^ 点 e E^ is called a (^) particular solution

b Solve

⼆ tcoslt 40 ⼆^1

The general solution is

ㄅ

ftcosctldttc

Q How^ do^ we (^) integrate this u t^ du

dcosctldtrtdv

tsinltl sinlt lg

fsi.lt dt (^) C tsinlt tcosct c

A htt

At tin

the slope of the tangent line to the

solution cure must be

m㼭 g 八 fiiiiii r.rs t You (^) can follow the^ lines to^ find^ the^ soluhbn

By attaching^ a^ short^ line^ segment As^ dope

Me (^) fit D at t.gl we^ obtain^ the slopefield^ of^ the^ ode t.eflt.DE

Here are some examples on the screen i

0 笨 ty 2

dy

0 不 ⼆ TG s^ ⼼^ O

ㄅ t o is a solution

dy

Es

⼆ dtfcidy

fdttc

12ci ttc⼆.^ sk主 trio Set (^) 芊 is^ also^ a^ solution ODE existence (^1) uniqueness

Theorem

let (^) flx.DK a function^ such^ that Dyflxin is Cont (^) on some (^) Rectangle containing Ito^ ⼼ Then (^) for some (^) open interval (^) of to in^ R^ the^ IUP