Unit Step Functions and Translation in T

Explore the intricacies of unit step functions and translation in t, focusing on Laplace transforms and their applications. Delve into the Laplace transform of unit step and rectangular window functions, as well as shifted unit step functions. Examine graphical representations and transformations of various functions, including t^2, (t-4)^2u(t-4), and t^2u(t-4). Compare translation in s versus translation in t, and practice solving inverse Laplace transforms involving unit step functions. Apply these concepts to solve differential equations using Laplace transforms with unit step functions, providing a comprehensive understanding of Section 7.6 from Nagle's textbook.

Laplace transform of unit step function & rectangular window function.
Laplace transform of f(t-a)u(t-a), the shifted unit step function.
Graphs of t^2, (t-4)^2u(t-4), t^2u(t-4).
Laplace transform of t^2, (t-4)^2u(t-4), t^2u(t-4).
Inverse laplace transform, translation in s vs translation in t.
Laplace transform of unit step function, sect7 6#1.
Inverse Laplace transform with unit step function, sect7.6#13.
Inverse Laplace Transform with unit step function, sect7.6#15.
Solve differential equation with Laplace Transform involving unit step function.