0ctl5,2o19_Last time

constant coefficient 2ndorder 4in DE

an t bu t cu 0rt

U e plr ar't br t c O CharacteristicEquationTwo real district roots

U er't u erstTwo complex roots r atif

u edtcospt ni edtsiupt

Repealed root real

u ert vii tertWas derived by assuming Ur _u.lt t solved furt Method of Reduction of Order

Spltidtt ftp.E

Applicable to general equations of the form

U t plan't911u 0

NexthopicThe inhomogeneous equation

u t pix u t glau gladThe solution will be composed of two pieces

me Uh 1Upw usolves g o solves geoParticular solution

EI n 2xu XXL

Sol to u 2 0 is ult enE Sol to u 2xu x is mix I particular solutionP

next ce Iif u U are the fundamental solution set to

U t plan't911 u O then any solution toU tplan t qlxlu glad must be of the firm

UK calx t Ciudad 411wparticular solution

Lemaire If U V solve 11 then w u v must solvethe homogeneous eguntin Lu 0

Proof Lw Ltu ul Ln Lv g g O

Example u tu 2xHomogeneous solution are u cosX M SuiX

And clearly up 2xMhd c cosy CrainX t 2x

This is great but how do we find the particular solutions

Method of Variation of ParameterstextbookLet the solutions to Ly 0 be Yi 192 notatcoin

Ansett yplxt uilxly.li tUzhdyzlxl YlxI 9unknown fuckinthe varying parameters

idea with ymyr known the equator Ly g is onlyoneeguatwin for the two unknowns u Uz Let US choosethe other condition so that things simplify

rf U y t Ury u4iuiy.tuiyituiyztuiyiluiyitu.ge Luigituiya

II u y Miya 0 then 4 contains uo 2udo.de derivatusof n U This is one additional constraint equation usedto determine u U

Compute 14 using this constraintLy y t play't 9114

mini tiny tuiyituryi t pix uigituigiglad my mya

Mily pig tgyi truly tpyitgy.ieniy tuiy

Mitty tutty tu y triguigituigi

So Mi Uz must satisfy

wig turfy 0nig twig g in order for Ly g

is 3 L Hit

Thesolution can be obtained by Gaussian Eliminationn g yr Nj 991y.yi y.ly y yi yiya992 91Wlyy4 wly.ir

And u u can be obtained by integratingthese expressions

Eixample

2g 3g t y e x41 exconstant coefficients

LySolve using characteristic equation

2r 3r 1 0

r 31 94.274

3 I Yz

y.hn ex ydxI e

By the previous variation of parameter calculation we have thatWly y4 y y y y

ft 1 e e e2

g x't 1 e'tU x 1 exe 2 X't I

e312

u 233 2x

U x't1 exex