LASTTIMEMCxy dx

t Nlx y dy o exact if My _Nx

integrating factor

if MJjN depends mly can find fucx set 4 1451N

then µ Mdx lundy is exact

if MYn depends oily can find lucy sit I MymN

Secondorder linear homogeneous equations

IVP for an nth order ODE

a Giddy tanCxid E ta dddxtaocxly

gcxsycxo7yo.ly Cxo7 Y yd xo Yu i

th existence anduniqueness

Let aix Adn g becontinuous or an interval I and let a Cx 40forevery x c I If XoC I then a solution yan of the IVP exists or Iend is unique

2y tDy'tGy 0solute existsand is unique on I C as as

y67 0 y 67 0y o is a sol it is theonly soluther

pointswhere 2 is desert orwhere sin x o

E sin g t y I x

T O 1 2 IT

y D o y Ct 1 TxSo Sol exists andis unique for XC 0,2

e nthorder ODE of the form

a G DI tanCxd ta G DI aocxly OIx din i DX

is said to be homogeneous

The superposition principle 7

Let y ya be solution of a homog ODE on an interval I Then

thelinear combination y C y G t t Ckyucx where Ci Cia are

arbitrary a stents is also a solution on I

Corolla a if y CH is a solution of C a constantmultiple Cy Cxis also a solution

b a honey OD E alwayspossesses the solution y o

Ex y y o has solutions y ex y e

Thus y C ex Cre is a solution forany G a

we are interested in a l yi t set ofsolutions y yn3 of

dei Lett f x fnCx befunction possessing at least n i derivatives each

The determinant wtf fa

it called theWronski of the functions

The criterion for Eu indep solutions

Let ya Tn be n solutions of nthorder lie honey ODE on an interval I

Thentheset of solutions is kindependent on I iff W y ya 0

forevery E I

In fact W y yn is either zero everywhere or I or nemero everywhereon I

Abe m if y yn solutions of anCxly ta Gy t tao 9 0

then WCy y C eS dx for some constant C

y Sin x y _as x two solutions of y ty o

www.t In I I I 1 1 toforany

So Yi Yee ane li independent

DEI Anyset y y of n en indep solutions ofn'thorder1in homeyODEHon an interval I is said to be a fundamental setof solutions on I

The a fundset of solutions of C exists on an interval I Cohere a Saccenta on40

The Let ya syn be a fund set ofsol of C or IThen thegeneralsolut of Cx on I is

y C y G t t Cnyncx where Ca a are arbitraryconstants

Ex y X't y X two sols of zx y y o on I Costas

Wly yr I k II X K Ix Ks Zx K fon

y y a kind set of solutionsA

Thusy C X ke ax generalsolution of a Costas

Reduction oforder s 4.27

2ndorder honey Cnn ODE aix y ta G y t a G y o I

suppose we knew one solution yr Wewantto look for a second linearlyndep.toy

Solution y as yzC uCx y Cx Substituting y hey in C 7wefind u

E y y D y e as a solution on C as as Usereductionof

ordertofind a second sol y

Soli y uCx7y Ucx ex y u e t y e

y u e t u'e tu extueu e't z u e't u e

y y ex u tzu o

U tzu o 7 W t 20 0 e2 w oe denote U w linear1stordereq

htfactor µ e22xw C e or U C e H

uU C e t a y ay c e t Cee I

choose C 2 C O Yz e

ble e fee 1 2 to ly yi fend.se ofioht6

is actually a general solution of y y o on C co is

Participationquest

Is sinly dx t Xcosly dy o an exact equation

SI Mexy sing My yI cosyNCx y x cosy Nx Off cosy

thus My Nx the equation is exact

in fact ftp.x.IM is solved by f x say