Use the Laplace transform table and the linearity of the Laplace transform to determine the following transform. Lleft{e^{3t}sin(4t)-t^{4}+e^{t}right}

Solving a differential equation by power series.
I want to find the power series solutions about the origin of two linearly independent solutions of
${\displaystyle w-zw=0}$.
Also, how do I show that these solutions are analytic?

Transform:
${\displaystyle f\left(t\right)={\cos }^{2}at}$

Differentiate.
${\displaystyle H\left(u\right)=(u-\sqrt{u})(u+\sqrt{u})}$

Solve the initial value problem using Laplace transforms.
$y"+y=f(t),y(0)=0,y^{\prime }(0)=1$
Here
$f(t)=\{\begin{array}{ll}0& 0\le t<3\pi \\ 1& t\ge 3\pi \end{array}$

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.

You drink a Starbucks Coffee in SM Seaside with a 120 mg. of caffeine. Each hour, the caffeine in your body system reduces by about 12 percent. How long will you have a 10 mg. of caffeine?

A metal is heated up to a temperature of ${\displaystyle {500}^{\circ }C}$. It is then exposed to a temperature of ${\displaystyle {38}^{\circ }C}$. After 2 minutes, the temperature of the metal becomes ${\displaystyle {190}^{\circ }C}$. When will the temperature be ${\displaystyle {100}^{\circ }C}$?