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

Solve initial value problem using laplace transform tables
$y^{\prime }=t+3,y(0)=2$

Solve the differential equation $(dP)/dt=4P+a$

We give linear equations y=-3x

Laplace transform which one is correct
$L\{t+e^{3t}\}$
1)$L\{t+e^{3t}\}=\frac{1}{s^2}+\frac{1}{s-3}$
2)$L\{t+e^{3t}\}=\frac{1}{s^2}-\frac{1}{s-3}$
3)$L\{t+e^{3t}\}=\frac{1}{s^2}+\frac{1}{s+3}$
4)$L\{t+e^{3t}\}=\frac{1}{s^2}-\frac{1}{s+3}$

Quickly! Need help
Solve differential equation, subject to the given initial condition.
${\displaystyle x\frac{dy}{dx}+(1+x)y=3;y\left(4\right)=50}$

For a given differential equation
$x{y}^{\prime }+2y\log y-4x^{2}y=0;$
$y(1)=1$
I want to use the substitution $v=\log y$. Which implies that
$v^{\prime }=\frac{dv}{dy}\frac{dy}{dx}=\frac{1}{y}{y}^{\prime }$
Hence:
$y^{\prime }=y{v}^{\prime }$
That being said, solving the equation I try as follows:
$xy{v}^{\prime }+2yv-4x^{2}y=0$
$x{v}^{\prime }+2v=4x^2$
$v^{\prime }+\frac{2}{x}v=4x$
$v^{\prime }=4x-\frac{2}{x}v$
How can I use seperation of variables here? Or how can I solve this thing ?

${\displaystyle y\frac{dy}{dx}=e^x}$
Solve it by separating variables like this:
${\displaystyle ydy=e^{x}dx}$
${\displaystyle \int ydy=\int e^{x}dx}$
${\displaystyle \frac{y^2}{2}=e^x+c}$