Suppose that you want to solve the following linear, time-invariant, non-homogeneous ordinary differential equation of order $N$:
$$\sum_{n=0}^N a_n x^{(n)}(t) =k \delta(t),$$
where $a_n \in \mathbb{R}~\forall n$, $a_N \neq 0$, $x^{(n)}(t)$ is the $n-th$ derivative of $x(t)$ ($x^{(0)}(t) = x(t)$), $k \in \mathbb{R}$ and $\delta(t)$ is the Dirac delta. Suppose also that initial conditions are $x(0) = x^{(1)}(0) = \ldots = x^{(N-1)}(0) = 0.$
I am aware that the problem can be easily solved using the Laplace transform.
Anyway, after a lot of ODEs solved, I realized (???) that the problem can be stated in a different way, leading to the same solution. My claim is that we can express the problem as an homogeneous one:
$$\sum_{n=0}^N a_n x^{(n)}(t) =0,$$
and the initial conditions $x^{(0)}$, $x^{(1)}$, $\ldots x^{(N-1)}$ depend on $k$.
Is this true? If so, how can we formalize this?