Question

4. Solve the IVP y" + 4y = 36t² + 34t, y(0) = 0, y0) = 0 b) 4y" - y'= 4 + 122, y(0) = 0, 7(0) = 0, y"0) = 0

Answer 1

y"+qy' = 3642 +3 at y lolzo Y(0) So the characterstic Creation is gut qy'=o 2 +40=o o D(0+4)=0 DEO, - complementy solution Y c = at a e at find the particular solution of the differen Hoo equation yp = 30 +2 +394 D²+ 90

9р - 342 424 о (не) OPE " ( 2 ) 1 ст за 26E1 11q Эр- І, 1 - D4 D 36 +1+34 +1 to takes ) ! (at 4.4 16 p = 3, E414 - 16+ – 65 145 Р: 1 C262 - (st - 1 P + ( 4 4 + 1 * at ] UP 9, 14 2 2 - 1 HP - Дt 1 - 2 2 - 6

The complete solution al chifferential eguation y = yetup y = atzet + 3t²-24²_t Apply the boundary condition as to y=0 o=ata - G=-62 APRly the second toy'zo boundary condition at Q2e +92-ct- 0= -4c2 torol = G = -1/ [c1=-62 = 1 then the complete solchon is. y=1 / 2 - 1 é + 3+² 22²_1 Y = 0.25 (1-ert) + at -2+2-t A

qy" - y'e at +12624 y(0) 0 7 (0)=0 y" (o)=0 The characterstic @geahon is 40²-0=0 D(?p2-03 59 0-0, 40-70 D² = 19 7 D= I12 complimenty UUUUUUUUUUUUUUUUUUUUUUUU Solution ? u Ye = at gent 0.5 + 0.5 t) The particular solution of the differential egreaton is. Jp = 4 + 120 2t 4D²-p

Yo = qt + 12 e 2t 9.D ² P yp- 4 + 12e2t tt 12024 D(40²-1) -D.(1-482) Y p = (1-2025 ( 44 +12224) yoon+ (1+202) (94 +1264) ype 44712824 + 192 et ] ypa ol la nit 12 (4 + 192 kq 1 Yp z a(2+2 +682 + 96 e 24 (Ypa - [244 770224

Then the saletion of o the the differ differential care 9 - 9 + 9р 9= 0.5t -0.5t Citae tge a 2t²_102e21 apply the boundary cs +=0, 9=0 condition o=4tc2t - O Рpe 2 ч. the second Apply the to y=0 boundan condition UUUUUUUUUUUUUUUUUUUUUUUU ♡ = 0 +0.562 8" -0. 5t 0.56e 2 a at-zote o= 0.502.-0.50 = G=G - 6 Ao third bound any condition al toyl=0 j" - 0.5t 0.25c est +0.25.6-0.57 - 2t a - Toge

o= 0.25 G +0.25 - 4-208 0.25 (2 to 250g= a12 czteaz 162€ - ☺ C2 teg= 1696 Solac eg © e 262 = 1646 2 = 2 = 829) C2 = 624 from eg o - Citata o C2-C C2+69] = - 824-629 a= -1648 Thon complete solution is -0-50 2 gz -1648 +624 oht +629 e - 24² - loze