Find the general solution of given differential equations

$y^{\text{'}\text{'}}+4y^{\text{'}}+5y=e^{-2x}secx$

$y^{\text{'}\text{'}}+3y^{\text{'}}+2y=\frac{1}{1+e^x}$

$x^2{y}^{\text{'}\text{'}}+x{y}^{\text{'}}-4y=0$

$x^2{y}^{\text{'}\text{'}}-4x{y}^{\text{'}}+6y=4x-4$

Find thesolution of given initial value problem

$5.y^{\text{'}\text{'}}-y^{\text{'}}-12y=0,y(0)=3,y^{\text{'}}(0)=5$

$6.y^{\text{'}\text{'}}+7y^{\text{'}}+10y=0,y(0)=4,y^{\text{'}}(0)=2$

Find the Laplace transform of

$7.t^{2}cosbt$

$8.t^{3}sinbt$

$9.t^4{e}^{at}$

Use Laplace transfer to solve each IVP

$10.y^{\text{'}\text{'}}+y=sin2t,y(0)=0,y^{\text{'}}(0)=1$

$11.y^{\text{'}\text{'}}-2y^{\text{'}}+2y=cost,y(0)=1,y^{\text{'}}(0)=0$