Use the appropriate algebra and Table of Laplace's Transform to find the given inverse Laplace transform. L^{-1}left{frac{1}{(s-1)^2}-frac{120}{(s+3)^6}right}

Solve for the following problems using substitution suggested by the equation. ${\displaystyle (3x-2y+1)dx+(3x-2y+3)dy=0}$

a) Find the Laplace transformation of the following function by using definition of Laplace transformation:
$f(t)=\cos (pt)\text{ where }t\ge 0$

Find Laplace transforms of ${\displaystyle \sin h3t}$ ${\displaystyle {\cos }^{2}2t}$

Find the Laplace transformation (evaluating the improper integral that defines this transformation) of the real valued function f(t) of the real variable t>0. (Assume the parameter s appearing in the Laplace transformation, as a real variable).
$f\left(t\right)=2t^2-4\cosh \left(3t\right)+e^{t^2}$

Find y" by implicit differentiation.
${\displaystyle x^2+4y^2=4}$

As part of trying to solve a differential equation using Laplace transforms, I have the fraction ${\displaystyle \frac{-10s}{(s^2+2)(s^2+1)}}$ which I am trying to perform partial fraction decomposition on so that I can do a inverse Laplace transform. When I try to work out this fraction, I get that ${\displaystyle -10=0(A+B)}$ since ${\displaystyle s^1}$ does not show up in the fraction. How does partial fraction decomposition work for a fraction like this?

Determine the Laplace transform of the following functions.
${\displaystyle \frac{2}{t}\sin 3at}$ where a is any positive constant and s>a