determine the Laplace transform of f. f{{left({t}right)}}={3}{t}{e}^{ -{{t}}}

find inverse Laplace transform of the given Function
${\displaystyle L^{-1}\frac{s-5}{{(s-5)}^2-9}}$

to find the inverse Laplace transform of the given function.
${\displaystyle F\left(s\right)=\frac{2^{n+1}n!}{s^{n+1}}}$

Let f(t) be a function on ${\displaystyle [0,\mathrm{\infty })}$. The Laplace transform of fis the function F defined by the integral ${\displaystyle F\left(s\right)={\int }_0^{\mathrm{\infty }}{e}^{-st}f\left(t\right)dt}$ . Use this definition to determine the Laplace transform of the following function.
${\displaystyle f\left(t\right)=\{\begin{array}{cc}1-t& 0<t<1\\ 0& 1<t\end{array}}$

Assume that the rate of evaporation for water is proportional to the initial amount of water in a bowl. If Jill places 1 cup of water outside and notices that there is ${\displaystyle \frac{1}{2}}$ cup of water 6 hours later. Can she use a differential equation to find when there is no water left?

Exact Differential Equations Solve ${\displaystyle (2x+\sin y)dx+(x\cos y+4y^3)dy=0}$

State the order of the partial differential equation ${\displaystyle u_{\times }+u_y=u+4}$. Then classify whether it is elliptic, hyperbolic or parabolic.

Explicit differential equations
Find all solutions of differential equation
${\displaystyle y^{\prime 2}-(x+y)y^{\prime }+xy=0}$?