Question

a,b, and c

1. Determine if the following function is a solution of a differential equation. a) y = 4e2' + Be for 6y"- 7y3y = 0 (3pts) b) y = Ae* + Be*** + 2 sin (3x) for y"+y'-6y=6cos(3x) – 30sin (3x) (3pts) c) y = Ae?" + Bx? for y"- 3y + 2y = 0 (3pts)

Answer 1

ļ ay y net bet We Differentiate worot. Se get y' = 2 2 4 3 & Bear Again differentiate worto 'x' wo get Be 24 103 Greven differential ear in 6y" – 7y' – 34 - 1x -0. 6 $(34218, **) - offwr.4.3 - 3 (4.172 Bell) = 0 --5) 42**(13-)*** =0 zn - ²x O. Be Ooté - O oto 0 = 0 is solution. + Be Hence, y= Apk

-3 we get -27 6 Cos(3x) Wort. x -32 18 Stu3x y = A 6²% + Bé +2 Sil (32 Differentiate Worot. x y' = 84e22 – 9 Be Ageun Differentiate we get y" = 4 Ae2x + g Be² geven differential can yut y'_by = 6 cos(33) – 30 Ster (32) 18inox) +(24e27-33232 +6 00S (34) -6(Ae?? +Bet+2 514 3x) = 6 cas (32) - 30 Sime) ht Now, put in govern -34 4A 027+ g be (4+2-6) A e27 + (9-3-6) Be 34 € 18 +12) Sim 3x + 6 cos3x = 6 Cos 3x - 30 SIM (32) 6 Cossu * -32 Oo Ae2? + Oo Be - 30 Souzr + Cassa -30 silu 3x 6 Cos 3x - 30 SIM 32 = 6COS 3x - 30 Shu 3y. L.HIS R.H.S Hence, +2 Sim (3x) is the Solution y = A e 2 + Be-3x

war get y = 40?? + B x2 Differentiate woroto a y' = 24227 + 2B2 Again Differentiate coort. 'a' noo get Y" = 4 422X + 23 I'n differential Now guren put there equation. y"-3y' +2y =o 44e2% + 25 – 3 (24227+23x) +2(4e294 B7)=0 (4-6 +2) AC2X + 2B-6BX +2B*2=0. 0.4223 + 2Bx? -6 Bx +213 CO. 2Bx?- 6 BX+2B = 0. Lites. + R.H.S. Hence, 27 + Bx² is fe 22 not solution A y =