Plan for Today: 3.1 Learning goals: Concepts: Linearly Independent Solutions, Wronskian, Characteristic Equation Skills:

Be able to find the solution to an IVP for a linear 2nd order ODE given two linearly independent 1.solutions for the ODE Be able to check if two solutions are linearly independent using the Wronskian 2.

Reminders-Announcements

Read the textbook! 1.Office hours today and tomorrow 2.Quiz 2 grades posted. 3.

Letty y p y iqcx1yo

Saw

y t y O

i u

can be solved by setting y ai czcosand finding a c T T

what makes themgood building blocks

Can we use ayytwo solutions of a linear

2nd order ein as building blocks to produce

guy solution

Ex y sin x Jr 3 sin

Both solve yty 0

Can we solve d as a linear comb ofSisi

c y E targetay II tug 1 2

C Ez g 3

E a 32 2nT't

such a.cz

Defir 2 functions f fz on an interval Iam called linearly independent ifc fault czf x 0 on I a cz o

T linear combinationwhere c ez constants

extent None of them is a const multiple ofthe other

If not linearly indep linearly dependent

Chek SiuGd cos liu indep bec

none is const malt ofthe other

sink 3sincx Lin dependent

Tumi Let y yz be two Cy independemsols of the homogeneous

y tip y tqGy o

p x qGd cont on an interval I

If y is as soin to then thereare a.cz such that

ya cry czy.CH

Any 2 liu indep sols are good buildingblocks

Soles of form a 2 dinic vector space

any 2 liu indep sols are a basis

E Ey Xy y O

y x

Yz xlnsols on I co o

y ya liu independent none of themconst multiple of the other

Find Solin of w ya 2 y417 3write y

c y Czyz find Ci Cz

Y G yallg l c I Cz O 2

Y l Cc h Cz I 3ya'll ye'Cxk lux I

Find Cz Ic 2

A way to check linear independence

wrouskian determinant

Eep Y yz sols of y t p y taek y 0

y cry Czyzwhen does y solve the 1UP yCal b

yCal bz

Try to find a.cz

gla b c y a tczyz.la b

Y a bz c y ca tczyical beknown

I c X Lunmown

can find a uniquedef bilal y.ca

Pair of G.cz exactly when

y ia0

Wronskiour determinant

Why yz defY CH Yz CH

Y CH ydCH

y yi yay

THI I f y yz sots to

y t play'tqlx1y 0

on I p q cont

y yz live.dep.eu swCy yz 0 everywhereon I

y yz lineup out Whyyz f 0 everywhere

on I

E W cos sink costed SinhaT P

sols of y t g 0 sink cost

cos tsin Lnever 0

cos x sina liu indep

W X xenx X xluxt xsols

x2y Xy y O L lux 11

Klux x xluxX f O ou 0 D

Tf The W of two lice independent functionsnot solutions of Hee save egh can vanish

only at one pt

we x sina.it

Xcosed sinkw x sincx o 0 but x Siu

liu indep ou too o

but they are not sols ofthe same 2nd order linearcoin no contradiction

Ryle If g c ya so ya yehot lice inden

W y yz YIN cycad

yik cyicxl

cyicxsyzkxl cyicxly.ec0

how to see that liu dependencewrouskian O