Use the Laplace transform table and the linearity of the Laplace transform to determine the following transform Lleft{3e^{-4t}-t^{2}+6t-9right}

How to find f(x) if ${\displaystyle \frac{df\left(x\right)}{dx}=f\left(x\right)}$? I know ${\displaystyle c{e}^x}$ is a solution, but how does one find it and how to prove it is the complete solution?

Determine $L^{-1}\{F\}$
${\displaystyle sF\left(s\right)-2F\left(s\right)=\frac{2s+2}{s^2+4s+4}}$

Radium decomposes at a rate proportional to the amount present. In 100 years, 100 mg of radium decompose to 96mg. How many mg will be left after another 100 years What is the "half life" (the time required to decompose half the initial amount) of radium?

Find the Inverse Laplace Transform f(t) of the following if F(s) equals:
${\displaystyle \frac{1}{s^4-4s^2}}$

Find the laplace transform of
${\displaystyle F\left(s\right)=\frac{11s+28}{{(s+2)}^2(s+5)}\to f\left(t\right)=L^{-1}\left[F\left(s\right)\right]=?}$

How to solve ${\displaystyle a\ddot{u}+b{\left(\dot{u}\right)}^2+\dot{u}+\dot{u}c{e}^u+e^u-e^{2u}+1=0}$