Question

Find the general solution to the following differentiel equations USING VARIATION OF PARAMETER METHOD.

. y'"' + 4y' = t y(0) = y'(0) = 0 et y'(0) = 1

Answer 1

If Yi T 7 is the solution of The coveresponding homogeneous differential equation of sind an der Then W[n) = fr 72" Yg 7, W2 = W3 Y 9 Then If Perticular integer is Yo = Uilot ay UzYz Then ' F اليا Pow " Wr f PW F pu U where P. (M) y" & (P) y" & 12) 4' & P (2) Y = F (M) differential equation. is the

Here + the differential equation t, (i) Po(t) = 1, F(t) = t The Auniallary equation m² x 4m 0 7 m = +2i, complementary function corresponding homogeneous equation) Got cosat & Casimat here, 7,1A) = 1 72 (t) = corset, (t) = sinat Then ( solation of sim at cusat - 2 sim 2t wit) = 0 8 2 cost - 4 sinet 0 -4 caset cost strat Wild) = -2 sinat 2 eint Wyrt) = sinat a cost - 2 cos at O و (4)دلا cost - 2 sinat 2shat F Pow t 1.8 ht 2 ने t. 2 " 3 t. 2 costa 7 teoszt

1 & Et) = s & cost de [ taking t'as ist & 'eost as second function using integration by Parts] sonat dt t - sinat 2 1. 52 t sinat - 2 ) Simat de ric Coszt sinat [ & dan ] t sinat cosat 16 $us (95-2+) 4. (t. sinat) -- Uz (t) = 1 & sinat de t taking t'as first & sinat and function, using integration by parts, we get -Coszt cuszt * 1 t. 2 2 C- cost & cost at 2 2 - :-4 [- + cos2t + 2 Staat udz 2

Uy(t) = & count to sinat imperticular let d = d = 2 = 0] Then Ip - & te 1 + (- t sinator -76 cos2t) cost To sinre) since t coszt 8 8 t sinateosat - to I costat & costsinat to sin at 16 of ů) i The solution Alt) = C, a cost sinatt c 8 Given that 70)=0 tego to = 0 -> Ctr. To Now, from ) y) 2623 zsinat & 2 eus 2t & given 4/10)=0 20₂=0 = 0 A 4 (13)

Again, from (iii) Y"(t) = -4 C Cosat - 4 Cg sina + 2 ON given, &"(0) = => 1 = – 4 C 24 => -46 2 C = - 3 / 3 (C) from G = to * :. 424 425-16 = 0 . The The required solution of solution of the giren " + 4y = t with it with y) = y(0) 20 4 7" (0) = 1 Y(t) = 7 - Po cusat & + 2 I using variation of Parameter method] (M) 16 ) Y 8
since no question is mentioned, as per rules the first question is answered. With the same process we can solve the others too.