mat 3724 applied analysis i
DESCRIPTION
MAT 3724 Applied Analysis I. 1.0 Review. http://myhome.spu.edu/lauw. Teams. 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 - PowerPoint PPT PresentationTRANSCRIPT
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 xydx
dyx
2
25
w w
t x
Linear (First Order) D.E.
)()( xfyxPdx
dy : 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 e
dx
( ) 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 y
dye e y e
dx
( ) 3
( ) 3
( ) 3
P x dx x
P x
P x dx x
e e
Example 2
0)9( 2 xydx
dyx
**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
A
x-x
xedxxP
:
)33 i.e. 09 (Assume
?9 choose weif What :Q2
2)(
Remarks0)9( 2 xy
dx
dyx
The Chain Rule
dx
du
du
dy
dx
dy
xgfy
xguufy
))((Therefore,
)( ),(
dy
dudy
dxd
y
u
x
u
dx
The Chain Rule: Case 1
, , , z f x y x g t y h t
dz
dt
z
t
y
z
x
z
y
dx
dt
x
dy
dt
Example 3
Find
2 2ln , sin , tz x y x t y e
dz
dt
z
t
y
z
x
z
y
dx
dt
x
dy
dt
The Chain Rule: Case 2 , , , , ,z f x y x g s t y h s t
z
s
z
t
z
t
yx
tss
z
x
z
y
x
s
x
t
y
s
y
t
Example 4
Find
2 2ln , sin , stz x y x s t y e
z
s
z
t
yx
tss
z
x
z
y
x
s
x
t
y
s
y
t
Other Cases
Similar
Example 5
Show that
is a solution of the Laplace’s Equation
2 2
2 20
u u
x 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 6
Find all the u(x,t) such that
5u
xtx
,u x t