\varphi^1 & \dots & \varphi^p \\
\varphi^1_1 & \dots & \varphi^p_1 \\
\vdots & \ddots & \vdots \\
\varphi^1_{p-1} & \dots & \varphi^p_{p-1}
\end{pmatrix}$$ for the corresponding Wronskian. Solutions $\varphi^1,\dots,\varphi^p$ of the homogeneous equation form a fundamental set if and only if the Wronskian $\mathrm W(\varphi^1,\dots,\varphi^p)$ does not vanish at $x_0\in (a,b)$, and therefore it vanishes nowhere on $(a,b)$ in view of the Liouville–Ostrogradski formula, $$\mathrm W(\varphi^1,\dots,\varphi^p)(x)=\mathrm W(\varphi^1,\dots,\varphi^p)(x_0)\exp\left(-\int_{x_0}^x\frac{g^{p-1}(x')}{g^p(x')}\,{\rm d}x'\right).$$ In this case, $(\varphi^1,\dots,\varphi^p)$ is a basis of the vector space of solutions to the homogeneous equation. See, e.g., [@Hartman2002 Section IV.8.iii] or [@Walter1998 Section 19.II]. A particular solution to the inhomogeneous equation is given by (cf. [@Amann1990 Proposition (14.3)]) $$\label{eq:particular_sol_from_wronskian}
u(x) = \sum_{s=1}^p (-1)^{p-s} \varphi^s(x) \int_{x_0}^x\psi^s(x')\,{\rm d}x'
\quad\mbox{with}\quad
\psi^s:=\frac{f}{g^p}\frac{\mathrm W(\varphi^1,\dots,\lefteqn{\varphi^s}\!\smash{\diagdown}\,,\dots,\varphi^p)}{\mathrm W(\varphi^1,\dots,\varphi^p)}.$$ Finally, we recall the well-known fact that any $p$th order equation of the form can be rewritten as a linear system of first-order ordinary differential equations in the normal Cauchy form, so that the solution theory of (scalar) equations of the form can be reduced to that of such systems. Concretely, setting $$v^s:=u_{s-1},\quad s=1,\dots,p,$$ equation is equivalent to the system $$\label{eq:ode_system_no_parameter}
v_1 = A(x)v + F(x)$$ for $v=(v^1,\dots,v^p)^{\mathsf T}$, where $$\label{eq:matrix_system}
A = \begin{pmatrix}
0 & 1 & 0 