Use Laplace transforms to solve the initial value problem x

Use Laplace transforms to solve the initial value problem x\" + 2x\' + 4x = 0, x(0) = 1, x\'(0) = 1

Solution

Taking laplace both the sides-

S²x(s)-Sx\'(0)-x(0)+2[Sx(s)-x(0)]+4x(s)=0

Putting given initial values

x(0)=1, x\'(0)=1

S²x(s)-S-1+2[Sx(s)-1]+4x(s)=0

S²x(s)+2Sx(s)+4x(s)-S-1-2=0

x(s)=(S+3)/(S²+2S+4)