Find the solution of initial value problem using Laplace tra

Find the solution of initial value problem using Laplace transform yPrime + 2yprime + y = 0, y(0) = 0, yprime(0) = 3

Solution

y\'\'+2y\'+y=0, y(0)=0, y\'(0)=3

L(y\'\'+2y\'+y)=L(0)=0

=> L(y\")+2L(y\')+L(y)=0

=> t²Y(t)-ty(0)-y\'(0)+2{tY(t)-y(0)}+Y(t)=0, L(y(x))=Y(t)

=> t²Y(t)-3+2tY(t)+Y(t)=0

=> (t²+2t+1)Y(t)=3

=> Y(t)= 3/(t+1)²

=> y= L^-1[3/(t+1)²]

= 3xe^-x