I am trying to take the laplace transform of \cos(t)u(t-\pi).

Classify the following differential equations as separable, homogeneous, parallel line, or exact. Explain briefly your answers. Then, solve each equation according to their classification. ${\displaystyle (2x-5y+1)dx-(5y-2x)dy=0}$

If ${\displaystyle y={\left(\frac{x}{x+1}\right)}^5}$, then ${\displaystyle \frac{dy}{dx}=}$

Solution of the following initial value problem using the Laplace transform
$y"+4y=4t$
$y(0)=1$
$y^{\prime }(0)=5$

To find:
The Laplace transform of $L[e^{-3t}{t}^4]$

I want to find the laplace inverse of
${\displaystyle s^{-\frac{3}{2}}}$
the steps given in the solution manual are as follows:
${\displaystyle \frac{2}{\sqrt{\pi }}\frac{\sqrt{\pi }}{2s^{\frac{3}{2}}}=2\sqrt{\frac{t}{\pi }}}$
I know the first part ${\displaystyle \frac{2}{\sqrt{\pi }}}$ is obtained using the gamma function ${\displaystyle \mathrm{\Gamma }\left(\frac{3}{2}\right)}$ , but not quite sure how the rest is obtained.

Evaluate this laplace transform of ${\displaystyle \frac{1-e^{-t}}{t}}$ and the integral exists according to wolfram

What is the differential equation of the orthogonal trajectories of the family of curves ${\displaystyle x^2-3xy-2y^2=C}$?