Solve the given differential equation $by$ using the Laplace transform, and check out the correctness

$of$ the obtained solution, using the initial conditions.

$a{y}^{\text{'}\text{'}}+10y^{\text{'}}=e^{-3t},y(0)=1,y^{\text{'}}(0)=2.$

$b{y}^{\text{'}\text{'}}+5y=1+t,y(0)=0,y^{\text{'}}(0)=-1.$