MAT 3724Applied Analysis I

1.0 Review

http://myhome.spu.edu/lauw

Teams

Ainsley Hiegel 1.3 Adam Hanson 1.6+2.3 Alora BourbonnieEllen Kim Lisa Goodhew Nathanael SleightTara Walker Rachel Murphy Kristian Rubesh

1.5Hannah Judd 1.8 Kathryn Yancey 1.9Robert Rendle Taylor Elzinga

Salvador Eng Deng Shelbie Davis

Paul Schale 1.7 Everan Chaffee 1.4Josh Tjelle Jamey FrykholmPatrick Maguire Ryan Salgado

HW Download HW 1.0

Preview Integrating Factor (Linear First Order

ODE) Multivariable Chain Rule Be sure to pay attention to expected

presentation.

ODE, PDE Ordinary Differential Equations

Partial Differential Equations

0)9( 2 xydxdyx

2

25w wt x

Linear (First Order) D.E.

)()( xfyxPdxdy

: FormStandard

Technique: Multiply both side by the integrating factor

dxxPe

)(

Example 1

3 6y y

( ) ( )dy P x y f xdx

Verify is the general solutions

Verify…xCey 32

3 6y y

Expectations When there are two groups of related

calculations, do not mix them up.

3 3 3

3 6

3 6x x x

y y

dye e y edx

( ) 3

( ) 3

( ) 3

P x dx x

P x

P x dx x

e e

Two-Column Format Integration by parts, substitutions, partial

fractions, …

3 3 3

3 6

3 6x x x

y ydye e y edx

( ) 3

( ) 3

( ) 3

P x dx x

P x

P x dx x

e e

Example 2

0)9( 2 xydxdyx

**Reminder: Tell Wai to specify the interval at the end. He usually does not remember. You can take one point off from him. Ha, ha!

( ) ( )dy P x y f xdx

interval largest the for Solve:Tradition

Ax-x

xedxxP

:)33 i.e. 09 (Assume

?9 choose weif What :Q2

2)(

Remarks0)9( 2 xy

dxdyx

The Chain Rule

dxdu

dudy

dxdy

xgfyxguufy

))((Therefore,)( ),( dy

dudydx

d

y

u

x

udx

The Chain Rule: Case 1

, , , z f x y x g t y h t

dzdt

z

t

y

zx

zy

dxdt

x

dydt

Example 3

Find

2 2ln , sin , tz x y x t y e

dzdt

z

t

y

zx

zy

dxdt

x

dydt

The Chain Rule: Case 2 , , , , ,z f x y x g s t y h s t

zs

zt

z

t

yx

tss

zx

zy

xs

xt

ys

yt

Example 4

Find

2 2ln , sin , stz x y x s t y e

zs

z

t

yx

tss

zx

zy

xs

xt

ys

yt

Other Cases Similar

Example 5

Show thatis a solution of the Laplace’s Equation

2 2

2 2 0u ux y

( , ) cosxu x y e y

Expectations Always start with one side and show that

it equals to the other side. Normally, it is easier to start with the side

with more complicated.

Example 6Find all the u(x,t) such that

5u xtx

,u x t