Exact Differential Equations Solve (2x + \sin y) dx +

Solve the following first order differential equations:
1. ${\displaystyle (1+x^2+y^2+x^2{y}^2)dy=y^{2}dx}$
2. ${\displaystyle e^{x}y\frac{dy}{dx}=e^{-y}+e^{3x-y}}$

Write down the qualitative form of the inverse Laplace transform of the following function. For each question first write down the poles of the function , X(s)
a) $X(s)=\frac{s+1}{(s+2)(s^2+2s+2)(s^2+4)}$
b) $X(s)=\frac{1}{(2s^2+8s+20)(s^2+2s+2)(s+8)}$
c) $X(s)=\frac{1}{s^2(s^2+2s+5)(s+3)}$

Evaluate the following differential equations:
Integrating Factor by Formula
${\displaystyle (2y^2+2y+4x^2)dx+(2xy+x)dy=0}$

Solve the differential equation$2y"+20y^{\prime }+51y=0,y(0)=2$
$y^{\prime }(0)=0$

Using (1) or (2), find L(f) if f(t) equals: ${\displaystyle {\sin }^2\omega t}$

Second derivative using implicit differentiation with respect to x of ${\displaystyle x=\sin y+\cos y}$
${\displaystyle 1=\cos y\frac{dy}{dx}-\sin y\frac{dy}{dx}}$
${\displaystyle 1=\frac{dy}{dx}(\cos y-\sin y)}$
${\displaystyle \frac{dy}{dx}=\frac{1}{\cos y-\sin y}}$

Trying to find the Laplace transform of
${\displaystyle \frac{\cos t}{t}}$
It comes out as infinity, but that doesn't make any sense.
Does this mean that this function doesn't have a Laplace transform or is something wrong here?