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