updated draft up to page 13 to include equation (13.29) and in the updates we have repositioned the step function in equation (9) so as to be more mathematically formal along with some relevant Laplace transforms and basic inverse Laplace transform included in (13.5) there is also an emphasis on initial condition of the overall response
https://foxtrotpaparomeo.wordpress.com/2024/09/14/1492024the-inductor-capacitor-filter-lcf-study-workout-notes/