How to solve the following question using Laplace Transform (IVP) ?

(1) y''-2y'-3y=t.e^t
y(0)=0, y'(0)=-2
Laplace Transform
s^2 Y(s) - s y(0) - y'(0) - 2 [ sY(s) - y(0)] - 3Y(s) = 1/(s-1)^2
(s^2-2s-3)Y(s) = 1/(s-1)^2 - 2
Y(s)= 1/[(s-1)^2(s+1)(s-3)] -2/[(s+1)(s-3)]
set 1/[(s-1)^2(s+1)(s-3)] -2/[(s+1)(s-3)] = A/(s-1)+B/(s-1)^2+C/(s+1)+D/(s-3)
1 - 2(s-1)^2 = A(s-1)(s+1)(s-3) + B(s+1)(s-3) + C(s-1)^2(s-3) + D(s-1)^2(s+1)
1-2(s^2-2s+1) = A(s^3-3s^2-s+3)+B(s^2-2s-3) + C(s^3-5s^2+7s-3) + D(s^3-s^2-s+1)
A+C+D=0
-2= - 3A+B-5C-D
4= - A-2B+7C-D
1-2=3A-3B-3C+D
A=0, B=-1/4, C=7/16, D= - 7/16
Y(s)= 1/[(s-1)^2(s+1)(s-3)] -2/[(s+1)(s-3)] = A/(s-1)+B/(s-1)^2+C/(s+1)+D/(s-3)
= - (1/4)/(s-1)^2 + (7/16)/(s+1) - (7/16)/(s-3)
y(t) = -(1/4) t e^t + (7/16) e^(-t) -(7/16) e^(3t) ......Ans

(2) y''+4y'=8+34cos(t)
y(0)=3, y'(0)=2
Laplace Transform
s^2 Y(s) - s y(0) - y'(0) + 4[s Y(s) - y(0)] = 8/s + 34s/(s^2+1)
(s^2+4s) Y(s) = 3s +2 +12 + 8/s + 34s/(s^2+1) = 3s + 14 + 8/s + 34s/(s^2+1)
Y(s) = 3/(s+4) + 14/[s(s+4)] + 8/[s^2(s+4)] + 34/[(s+4)(s^2+1)]
14/[s(s+4)=(7/2)[1/s-1/(s+4)]

8/[s^2(s+4)]=A/s+B/s^2+C/(s+4)
8=As(s+4)+B(s+4)+Cs^2
A+C=0
4A+B=0
4B=8
A=-1/2, B=2, C=1/2
8/[s^2(s+4)]= - (1/2)(1/s)+2/s^2+(1/2)/(s+4)

34/[(s+4)(s^2+1)]=D/(s+4)+(Es+F)/(s^2+1)
34=D(s^2+1)+(Es+F)(s+4)
D+E=0
4E+F=0 16E+4F=0
D+4F=34 -E+4F=34
D=2, E=-2, F=8
34/[(s+4)(s^2+1)]=2/(s+4)-(2s-8)/(s^2+1)

Y(s) = 3/(s+4) + (7/2)[1/s-1/(s+4)] - (1/2)(1/s)+2/s^2+(1/2)/(s+4) + 2/(s+4)-(2s-8)/(s^2+1)
= 3/s + 2/s^2 +2/(s+4) - (2s-8)/(s^2+1)
y(t) = 3 + 2t + 2e^(-4t) - 2 cos t + 8 sin t ......Ans


(3) y'''+y''=e^t + t + 1
y(0)=0, y'(0) = 0, y''(0)=0
Laplace Transform
s^3 Y(s) - s^2 y(0) - s y'(0) - y''(0) + s^2 Y(s) - s y(0) - y'(0) = 1/(s-1) + 1/s^2 + 1/s
s^2(s+1) Y(s) = 1/(s-1) + 1/s^2 + 1/s
Y(s) = 1/[s^2(s+1)(s-1)] + 1/[s^4(s+1)] + 1/[s^3(s+1)]
1/[s^2(s+1)(s-1)] = A/s+B/s^2+C/(s+1)+D/(s-1)
1=A(s^3-s)+B(s^2-1)+C(s^3-s^2)+D(s^3+s^2)
A+C+D=0
B-C+D=0
-A=0
-B=1
A=0, B= - 1, C= - 1/2, D=1/2
1/[s^2(s+1)(s-1)] = - 1/s^2 - (1/2)/(s+1) + (1/2)/(s-1)

1/[s^4(s+1)] = A/s+B/s^2+C/s^3+D/s^4+E/(s+1)
1=As^3(s+1)+Bs^2(s+1)+Cs(s+1)+D(s+1)+Es^4
A+E=0
A+B=0
B+C=0
C+D=0
D=1
A = -1, B = 1, C = - 1,D = 1, E = 1
1/[s^4(s+1)] = - 1/s + 1/s^2 - 1/s^3 + 1/s^4 + 1/(s+1)

1/[s^3(s+1)]=A/s+B/s^2+C/s^3+D/(s+1)
1 = A s^2(s+1)+B s(s+1)+C(s+1)+D s^3
A+D=0
A+B=0
B+C=0
C=1
A=1, B= - 1, C=1, D= - 1
1/[s^3(s+1)]=1/s - 1/s^2+1/s^3 - 1/(s+1)
Y(s) = 1/[s^2(s+1)(s-1)] + 1/[s^4(s+1)] + 1/[s^3(s+1)]
= - 1/s^2 - (1/2)/(s+1) + (1/2)/(s-1) - 1/s + 1/s^2 - 1/s^3 + 1/s^4 + 1/(s+1) +1/s - 1/s^2+1/s^3 - 1/(s+1)
= - 1/s^2 + 1/s^4 - (1/2)/(s+1) + (1/2)/(s-1)
y(t) = - t + (1/6) t^3 - (1/2) e^(-t) + (1/2) e^t .....Ans