I am having a bit of trouble with heavy side functions. One of the questions from a past exam paper I am looking at asks:

Use Laplace transforms to solve the initial value problem

subject to initial conditions

Here H is the Heaviside (unit step function). I am totally confused on how to go about this. If anyone could please show me step by step I would greatly appreciate it because my lecturers notes on this particular question don't really explain much.