Use the Laplace transform to solve the given initial-value problem {y}{''}+{2}{y}'+{y}={0},{y}{left({0}right)}={1},{y}'{left({0}right)}={1}

Find Laplace transform
$L\{(t-3{)}^{2}u(t-3)\}$

Use the definition of Laplace Transforms to show that:
${\displaystyle L\left\{t^n\right\}=\frac{n!}{s^{n+1}},n=1,2,3,\dots }$

Proof for law of complex exponents using only differential equation
I just read that an elegant proof exists that the law of exponents also holds for complex numbers (a,b,z all complex):
${\displaystyle e^{a+b}=e^a{e}^b}$,
which only uses the definition that
${\displaystyle y=e^{zt}}$
is solution to ${\displaystyle \frac{dy}{dt}=zy}$,
with initial condition ${\displaystyle y\left(0\right)=1}$, so in particular ${\displaystyle e^z=y\left(1\right)}$.
I can only find a proofs which use the trig-representation of complex numbers.

A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 24 ft/s. At what rate is his distance from second base decreasing when he is halfway to first base?

One property of Laplace transform can be expressed in terms of the inverse Laplace transform as $L^{-1}\left\{\frac{d^{n}F}{d{s}^n}\right\}(t)=(-t{)}^{n}f(t)$ where $f=L^{-1}\left\{F\right\}$. Use this equation to compute $L^{-1}\left\{F\right\}$
$F(s)=\arctan \frac{23}{s}$

use properties of the Laplace transform and the table of Laplace transforms to determine L[f]
$f(t)=2(t-5)u_5(t)$

Inverse Laplace Transform without using equations
I'm stuck with a question about Inverse Laplace Transform here, but the use of inverse laplace transform equation is forbidden.
${\displaystyle Y\left(s\right)=\frac{e^{-\pi s}}{s[{(s+1)}^2+1]}}$