plan for today: start 3 - purdue university
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Plan for Today: Start 3.1 Learning Goals:
Be able to recognize a 2nd order linear ODE. 1.Know the terminology: homogeneous/nonhomogeneous second order ODE (!! Unrelated to 2.homogeneous 1st order equations in §1.6!) What is the superposition principle? 3.What does the theorem of existence and uniqueness for IVPs for 2nd order linear ODEs say? 4.What do solutions to linear homogeneous 2nd order ODEs look like? 5.
Reminders/announcements: Quiz 2 closes in 1 hour HW 13 will be extended In the next few lessons, writing in green color will contain optional connections to linear algebra Ch 3
3 I 2nd order ODE
Until now reducible 2nd order ODE
acy y y 1 0 Ex
Gig y oy yy O
ex xy 3g 5
today linear 2nd order ODE
Acxly t BCxyy Cady f xw w w
x
e y t costly 3y 3hr4Non ex y t 35 0
Terminology if FCHIO in then is
called a homogeneous 2nd order linear ODE
unrelated to homogeneous TT f
If FG 0 then called non homogeneouslinhomogeneo
non homogeneous
e y t cos G y 139 0
homoy the homog edu associated to
Fromuow ow divide by Acx
y t pCxly gaily fad11 Ii I
BIA CIA F A
Seeu when solving reducible 2nd order eeis
there were 2 free parameters
y o yC y Gx 14
Expect 2 pieces of info needed to specifya Solh
What 2pieces of info are good what should an
appropriate WP look like to have existence
uniqueness
Badi Tgif yankocheck y Asincx solves WP for any A
y d sink yprescribing values at different locations mightnot give a unique soda
The goodiufo_Given y t play't qkay fewp g f cont on interval Ina C I b bz E R
then there is a unique Solin to
satisfying Scat bi
ya bz
and it is definedoualloficcompare w Ex Onterm in 1 S
AO called au IVP for 2ndorder linear eels
Recall 1st orbe linear there was a
formula for sol's integrating factor etc
what do sol's look likeI Superposition principleLinear Homogeoini y t pCHy't g y o
y e y O
f y ya are sols to homogy cointhen cry Czyz is also a Solin
p f linearc cz are constants
combination
In ex ye sink
Yz cos yz cos cos y
I 9 y 1 Czyz cisincx tacos Cx
c sink a cosCx
c sincx ez Kos X
Cc y 1Czyeso Gy t Czyz is a Solin
superposition pr says that the solutionsof linear homoy 2nd order ODE forma vector space
Sup pr if we have some sols we can
produce more
y t y O unique
g3 site gxists
Try to create this solely as a linearcomb of yesinG yz cos
yc sin tacos
what are the ca c for which9satisfies
IVP if any
yea c Ez a Ez I
y c cos Cz sink
y c Ez izz 2
c
So y Fz sin fz cos x is mySolin
Questions Can we produce airy soda toan 1UP from linear combinationsof a pair of Sol'sAre all pairs good enough