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