Question

Find the general solution of the differential equations taking into account the initial conditions using the parameter variation method:

. y'"' + 4y' = t y(0) = y'(0) = 0 et y"(0) = 1 yiv + 2y" + y = 3t+4 ; y(0) = y(0) = 0 et y"(0) = y''(0) = 1 y" – 3y" + 2y' =t+e' ; y(0) = 1; y'(0) = -set y" (0) 3 2

Answer 1

y"+y's y"0=1 1 De 33²+20 (alten) (uten) D(2000-2) -Xe+ old-sty ly 3 yo) = yloso, Auxiliary egh it m+4m=0 7 m/m²+4)=0 mlm tai]=0 M=0, 21,-01 Yo = q**t ses gut G&in2x Ye = a + G cosent casin 2x 32 yu+by'+y= 34+4 yoo yo) -0 ykos - y" (0)=1 Auxiltary @qm for Homogeneous part of te mt + zgn?tter1=0 - mitt8m²+1=0 => z?+22+1 co (puti zauf) = (2+13=0 29-1-1 mt=4 = (+ 1)2 = mati ii mei,1, +,- Ye = (9+421 colut ((2 + Chorina (3u+4.2 .. (1“) y"-24" +2y=ttet Y0=1,410=yo- quxiliary equation :- 3w²t2mzo 3 ml wm+2=0 m (M-2)(mt) = m=0,1,2 Le fons 9+ Geft ge?x applying flo in egr 40=1 atat (3=1 = 6e" +22 eu =y00) = c ta ce © y = Yp+ Le Go 97 Geeltgene 321 ren. y = -29&then to close the = -4G cosa 463 ginen YO=0 - او 4 + C2 = 0 yp= A 27ti (8²715 c2+2 (3 = -1 en y = ext4 Gen YMO 4+463 using at 863=- s! 4 -(34+4) benerel sou will be Y = (t Guy loput (C3+44) o=yo) = 23 sg= 1 = y"O) = -46 - - - - 19 =- aYc = 2 - Illosen is the couplementary th Ć (sayit=u) 33+ Sun $+96773 -263 = 2 2 - 1 + ( 7432 3149) P.I= yp= 1 1 2 = 1079 ..ca C2 = - 1 2 - 2x - 52 = -1 4 + 3 9=1-C2-C3=1-1+5 2 = 5/8 -14-58, 69=1, Cs=-57 .. General 301 96) = get up 4 - 919682 +68)out heath *- Lose to anno