Properties of Homogeneous Eps b o

we focus on Y t a Ct y t a Ct J 0

Notation Ltg Y t act Y t a Ct YFoperator a function on functions

L Linearity

Them Lcc y tqY c LCT to LCL

Proof

L GZ Czyz Cc Y TEY t a Cc Y tsYzt O_O 9 Yet Czyz

c Y t ca Y T o c Y t a czYt ao C Y t O_O Cz Yz

9 Yi ta Y tao Y t

Cz Y t Q Y too Yz

c LCT t Cz Lodz µ

2 Superposition

Them LCT e o LCT o LLGY.to Ya so IProof LG Y tea C LCT t LCL 0

11 11O 0

Me

3 General Solution

Thrm i If 2 Yes with Y f c Zzare Sol of LCT O

Then every solution yof Cy O is

yet c Y Ct t Ca Zz Ct

Remaok i nd Y with 2 f c Ya Loy o Loyaare called fundamental solutions

Example Hw superposition Propertya Y is Sol of y ta Y ta Y O 1

b Y is Sol of Y t a y tao Y coset

Then TIF 2cg

L Yity solves CL CF

L Titta LCT f Ya 0 tarot

2 Y tYz Solves 2 T

3 27 Solves 1 CT

L 29 2 LCT 2 O 0

4 232 Solves 2 CF

222 2 LEY 2 CosGE f CosGE

FE

Rc si If LCYh 0 and Lodz bCt

Then L Tract t e Tgct bct

StExample i Shou Y e Tze 02 are

fund Sols of

y y 6 Y o1

L I

Sec y Est f e e't Y

Est Est 341 g gSt

9 eSt

3 Est g Est

Q s 6 e3T

O 1

Show L Ya O

THE

Example i Y 3 is sole of Y 1 Y 6 Y 18

Fond infinitely many more sols LCY

Sol LCY L 3 3 t s 66 181 1We know from precious Fx Y E'T YEE are solutions of

t 421 264 0 1So 9 Satisfies

L 3 te e'tL 3 t c L EstL Y t C O

18 1 r

Also Y 3tcze

Y LC's sett3 t g L est

18 t GO18 1

Yet 3 t e E't q ettt.FI

Edo

Exempted Find the maximum domain where the Soc

of IVP is certain to exist

Ct c y SI y t 4Ct Y ICE ct 3

y 2 L Y 2 o

Sody 31

isY t 4 y t

E 3

a do b

e tThe 04 is defined on to If e

C Oo l v 1 3 V 3,00

The solution is defined for certain on

C00,1 or C 1 3 or 3,00

The IC is out to 2 C Cl 3

So Jett is certain to exist on

Dy C1ME