Consider an LTI system described by the following differential equation,
$$ \sum_{k=0}^{N}a_k\frac{d}{dt^k}y(t) = \sum_{k=0}^{M}b_k\frac{d}{dt^k}x(t) $$
With initial conditions,
$$ y(t)|_{t=0}, \left.\frac{d}{dt}y(t)\right|_{t=0}, \left.\frac{d^2}{dt^2}y(t)\right|_{t=0}, \cdots \left.\frac{d^{N-1}}{dt^{N-1}}y(t)\right|_{t=0} $$
In the Signals and Systems book by Simon Haykin, the following is the result of taking the unilateral Laplace transform of the above equation,
$$ A(s)Y(s) - C(s) = B(s)X(s) $$
Where, $C(s)$ results from the initial conditions of the system.
However, shouldn't the actual unilateral Laplace be the following?
$$ A(s)Y(s) - C(s) = B(s)X(s) - D(s) $$
Where, $D(s)$ is the contribution from the following terms,
$$ x(t)|_{t=0}, \left.\frac{d}{dt}x(t)\right|_{t=0}, \left.\frac{d^2}{dt^2}x(t)\right|_{t=0}, \cdots \left.\frac{d^{M-1}}{dt^{M-1}}x(t)\right|_{t=0} $$
There was nothing mentioned in the text about $x(t)$ and its behaviour at time $t=0$.