Solving initial value problem using Laplace transform

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Homework Statement
Please see below
Relevant Equations
Please see below
For this problem,
1717971751310.png

1717971761542.png

I'm confused how,
1717971790764.png

However, not sure how to go from here.
$$
f(t) = \begin{cases}
1 & \text{if } 0 \leq t < 10\\
0 & \text{if } t \geq 10

\end{cases}
$$
By definition Heavside function is,
$$
H(t - 10) = \begin{cases}
1 & \text{if } t < 10\\
0 & \text{if } t \geq 10


\end{cases}




1 - H(t - 10) = 1 - \begin{cases}
1 & \text{if } t < 10\\
0 & \text{if } t \geq 10


\end{cases}
$$
However, I don't know where to go from here. Does anybody please know?

Thanks for any help!
 
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