Solve the differential equation using Laplace transform of y''-3y'+2y=e^{3t} when y(0)=0 and y'(0)=0

Using Laplave Transform, evaluate the integro-differential equation
$y^{″}(x)+9y(x)=40e^x;y(0)=5,y^{\prime }(0)=-2$

$y^{\prime }-{\int }_0^{t}y(\tau )d\tau =4t{e}^{-t},y(0)=3$

A thermometer is taken from an inside room to the outside ,where the air temperature is ${\displaystyle 5^{\circ }F}$. After 1 minute the thermometer reads ${\displaystyle {55}^{\circ }F}$, and after 5 minutes it reads ${\displaystyle {30}^{\circ }F}$, what is the initial temperature of the inside room?

Solve ${\displaystyle y{}^{″}+s^{2}y=b\cos sx}$ where s and b are constants. I have tried undetermined coefficients, but it makes such a mess that I keep getting lost, I also tried variation of parameters. Really my issue is staying organized enough to get the solution, my solutions are always just a little off from the book.
I have solved the complementary homogenous equation, this gave me a solution with ${\displaystyle \cos sx}$ in it... So that's off limits in the particular solution. I looked for a solution in xsinsx and ${\displaystyle x\cos sx}$, but as I said, it was a mess.

Use Laplace transforms to solve the following initial value problem.
${\displaystyle x^{\left(3\right)}+x\text{-6x'=0, x(0)=0 , x'(0)=x}\left(0\right)=7}$
The solution is x(t)=?

Find the laplace transform of the following:
MULTIPLICATION BY POWER OF t
$g(t)=(t^2-3t+2)\sin (3t)$

I am stuck with this equation. If you can help me
${\displaystyle y{}^{″}\left(t\right)+12y^{\prime }\left(t\right)+32y\left(t\right)=32u\left(t\right)}$ with ${\displaystyle y\left(0\right)=y^{\prime }\left(0\right)=0}$
I found the laplace transform for y(t)
${\displaystyle Y\left(p\right)=x=\frac{32}{\left(p(p^2+12p+32)\right)}}$
so i need the laplace transform of y(t) and then the solution for y(t).