use laplace transform method to solve initial value problems

Question

y'' - 3y' - 4y =3 sin t , y(0) =y'(0) = 0

Answer 1

Step one : find the Laplace of both sides of the equation to be s²y(2)-sy(0)-y'(0)-3sy(s)+y(0)=3/(s²+1)

Step two: use give initial conditions and rearrange algebraically to find y(s)=3/(s(s2+1)(s-3))

Step three: use partial fractions to change y(s) = A/s+B/(s2+1) +C/(s-3)

Step four: find constants A,B,C using algebra by setting step 2= step 3

Step five: find Inverse Laplace of each of the three terms in step 3/4

Answer y(t)=-1-e^-3t+1/3sint