Question

USING THE PARAMETER VARIATION METHOD,

Find the general solution of the differential equations taking into account the initial conditions.

Note: only determine all the matrices W in relation to the particular answer Yp without calculating them

yiv + 2y" + y = 3t + 4 ; y(0) = y'(0) = 0 et y"(0) = y''(0) = 1

Answer 1

(ن) my yov + 2y" + y = 3+ ++ For C.F.; Put you + 2y"+y=0 and find Cifi solution e complementary function). Auxiliary equation is +2m²+1=0 m=11", 1 T yect) = ((i+ Cat) sint +(3 +(4+) cost -3) Now to find particular integral by the Method Vauiation of Parameters: The fundamental set 06 Solutions of complementary equation you + 2y " + 420 ; is "Y 11 % 21 % 3, 9 q } = { sint, tsint, cost, tcost} particular solution glutay" +y be Y p = 0, y,+ U2 Y2 + U3Y3 tuu ya Now. To determine We begin by imposing the following a conditions Ugg Uz guy! ui'y, + U2 Y2 + U3Yz + Uyyy 20 of 3t+4 let of (11) u Ug , U3, U4: ܕ ܙM } cs Scanned with CamScanner

1/-tiv) Since Agaiм iSwiki,4 с. : un 4 + 1,42 +33 +uv yu" 44 +u, 4, 4 и 93 + 4'' - 0 ч ч" + ч, 4,"vy" tu«Ч"-о Че -- ц, ч, tv, 4, tu; 93 + 4 ч 19 Р = ин ч ту, 4, +43 73 + ич 4 + uz, 1 ч, , tu} {3 +uu ач Иr - u, + ч, 4, +44 4 и 9 bctque uy tu, y2 тч; 9; +udч = 0 1 1) 11 ))) 1) E and Yp" Uy"+, 42" +43 +чччч Again , Differentiating this yields uiyiv ч, 4, + + ач 4 у ypl у iy V 4" + ч, ,"4 us' " + и чу Substituting the to d) we get u-41 +u, 242 + ;-43t uq 19ч - ич 13ч + Cu ." ц, ч," ч, Ч"ч''") Lч= yv4 24" + 4 1 3t+t (0) where CS Scanned with CamScanner

Ч»» » » Из » Чч ам. доlutіоnа o 4 | 24 so Lyi - О - ) We left with; u" + ч, ч," + "," "+ ч уч Зt + 4 combining this equation with equation in (iv) shows that ů a solution 4e - ui 91 + ч42 + 93 +ччч ө, d) 15 t ui'y t - Зt+4- и, че тч Чэ чу ч эр м, 4, 44, 4, + uz , uz ' =o ui y" + ч, ," + v;' уз" + чу'ч"> и "+ и, ч" + vз'93" чу чу"" which can be written in matrin form as Ч Я ч, 93" Уч" Ч" Уч" ч? Яз" Ч" Уч о 2. 3 ull б З 4. б (1) 1 ) 3 3t + Ч ज्व 11 чи" Я. cs Scanned with CamScanner

Determivaut of this «Не м «1. Ч. 2. » 03 , Чч is whanskian w of by cramen's Full ча Ligh j f Wi Е: 3:(+9 Po W fo - | (-yh-i (31 1ч) w и/ Те (3-4 ч) W, W u, (3t +ч), W — (1,", - 3 +-+Ч) W3 uy - (зE+ч) wч и И 45 Чу 41 43 94 Ч, w, - 2 Wy=yi — 9, ) Ч Ч Ч." Уч" , Ч" Ч. Чu" Ч Ч % 7 Ч Wз - ว Яз ч, 92 1 Wy= 9, ч. ॐ Уч Чу" 11 11 1) 4. " Я, Я" Ч" t , Y2 = tsint where y, = sint, yz = cast Yu- test sinu need to to calulate fuathu as per 110 we don't need А{e , to ие. Со dоu . CS Scanned with CamScanner

We are not supposed to solve it any further as per the note says that no need to calculate the matrices, so i am leaving the solution till here but without solving them we can't find the value of uj's and hence can't find particular integral and hence general solution. So to find the general solution first find out the determinants and then value of derivatives of uj's and by integrating find value of uj's and hence by putting them in yp find particular integral. And the general solution is y= yc+yp.