$$ y^{n}+p_1y^{(n-1)}+\dots +p_{n-1}y'+p_ny=g(t) $$
$$ L=D^n+p_1D^{n-1}+\dots p_{n-1}D+p_n $$
To solve the IVP for a $N_{th}$ order ODE, we require $n$ initial conditions $y(t_0),y'(t_0),...,y^{n-1}(t_0)$
If $p_1,p_2,...,p_n$ and $g$ are continuous on the open interval $I$, then there exists exactly one solution $y=\phi(t)$ of the ODE that satisfies the initial conditions where $t_0$ is any point in $I$. The solution will exist throughout this interval $I$
If $y_1,y_2,...,y_n$ are solutions of the homogenous equations, then the linear combination $c_1y_1+c_2y_2+\dots+c_ny_n$ is also a solution by linearity of the $L$ operator.
This set of solutions can express every solution of our homogenous equation if and only if it can satisfy all initial conditions:
I.E. the linear combination of solutions must satisfy this matrix
$\begin{bmatrix} y_1(t_0)&y_2(t_0)&\dots&y_n(t_0)\\ y_1'(t_0)&y_2'(t_0)&\dots &y_n'(t_0)\\ \vdots&\vdots&\ddots&\vdots\\ y_1^{(n-1)}(t_0) & y_2^{(n-1)}(t_0) & \dots &y_n^{(n-1)}(t_0) \end{bmatrix} \begin{bmatrix} c_1\\ c_2\\ \vdots\\ c_n \end{bmatrix}=\begin{bmatrix} 0\\ 0\\ \vdots\\ 0 \end{bmatrix}$
If we have less than $n$ solutions then we have an underdetermined system that cannot solve for all initial conditions, if we have more than $n$ solutions, we have an overdetermined system and we must have a solution that is linearly dependent of another.