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