Obtain the Laplace Transform of Lleft{e^{-2x}+4e^{-3x}right}

Solve both
a)using the integral definition , find the convolution
$f\ast g\text{ of }f(t)=\cos 2t,g(t)=e^t$
b) Using above answer , find the Laplace Transform of f*g

Consider the IVP ${\displaystyle y{y}^{\prime }=0}$ where ${\displaystyle y\left(0\right)=2\text{and}{y}^{\prime }\left(0\right)=3}$.Solve it using Laplace transforms

$y^{\prime }-{\int }_0^{t}y(\tau )d\tau =4t{e}^{-t},y(0)=3$

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

Find the Laplace Transform of cos h 3t cos 2t

Find the INVERSE Laplace transform: ${\displaystyle F\left(s\right)=\frac{2s+16}{s^2+4s+19}}$

The population of a certain country is known to increase at a rate proportional to the number of people presently living in the country. If after two years the population has doubled, and after three years the population is 20,000, find an expression for the approximate population of the country at any time and estimate the number of people initially living in the country.