Question

please do all 3.

Due Date 01/23/2020 1. So the allow Last Name First Nam differential equation 2y+ry'+y-3.1) - 1.5(1) - 2. 2. Find the Laplace t o m of the functie f) + + Sin (3) Compute C +

Answer 1

O zien cauchy - Euler differential equation is xy + xy + y = 3x, y) = 1, let x = et or etalning 911)=2 x2 - does not 22

d x2 d1 and more e py" - det putting values of xy, and sexyu un giren defferential equation we get dy + y = 36 D. day + y = 302t - characteristic equation of m² tl – o man met i

C.F. - Glastt) + C2 sin (D) And p. I of is poti = b (3 029) Deo = 3e2t - Bert then solution of equation ② is y = G cas (t) + (2 series + 7 ezt Now substituting talna in we get = G Cas (In (47) + (2 sin eng + z elax) - cas fln x) + (z sir I nx) + 3x²

due hate. y (1) then Yl) = Glas In 1) + (2 sistlul) + 3(1 = G cas (0) + (2 sen (0) + 2/3 = 1 G= t 53 y = -G sin Elna) + Cz cas Inx) & gt -Gsin cnx) + ( cas (In) f some ze re X2 9'11 =1 = + (-4 sir (0) + (2 cas (0) + 6(12) => 3.1+ = 1 =) = - Fire + Hene required solution is y= 2 Cas clix) - J serglan 3 2

lil. = +(2(as (lnx) - sin Clux) + 3x) (2 giren funtion is fel-t et +960 2t + 5 sus (30) taking Laplace transform both side we get L(fl) - L (d 2 ) +86 Leet +5 Lisin (341) + 583 F (A = 12, + 46 82 +9 (3-611/2+1 S-2 29 + 15 2 A+) 3 s -2 3279 which is required in of giren function Laplace transform

- 30-9 I 3 first of all we will use partial fraction 30 - 9 st - 6x +3 - (A- (3406)) (x - (3-16) 30-0 (2-3-J6) (0-3706) How let 30-9. – A ta (6-3 706) (P-3-6) (4-3-86) (4-37.6) 30-9 = A (s-3+6) +B(0-3-15) 1 = s(A+B) - 3 A+T6 A - 3B DEB A+B=3 -A (3-16) – B (3+5) = – 4 A (3-16) +B (3+85) = 4. 3(A+B) + 6(BA) = 4. g tro (B-A) = 4

How taking Laplace both side in above L+(35-9 s – 60 +3 Inverse equation transform. we get = ( 168 4. maj 2" (+6wva +(1.4-SVEJ 2--vo) (18+576) Etrole (10-516 ) ( 86 ) 1 12 herefore (3 -4 oz-6872) = PESAVU) F4 p&SVOIP whes L (8 + 556 12 lo- 5.55 12 ß-
formula used in calculating Laplace transform:

Formula! @ Leat) = sa ② L (sin (1981) = shfar n! = 12,3, +