given by the n equations dy/dx =
y1, dy1/dx = y2, ...,

dy_n-1/dx = p - p1 y_n-1 - ... - p_n y;

but in fact any simultaneous system of ordinary equations is reducible
to a system of the form

dx1/dt = [phi](tx1, ... x_n).

Simultaneous linear partial equations.

Complete systems of linear partial equations.

Jacobian systems.

Suppose we have k homogeneous linear partial equations of the first
order in n independent variables, the general equation being
a_[sigma]1 df/dx1 + ... + a_[sigma]n df/dx_n = 0, where [sigma] = 1,
... k, and that we desire to know whether the equations have common
solutions, and if so, how many. It is to be understood that the
equations are linearly independent, which implies that k <= n and not
every determinant of k rows and columns is identically zero in the
matrix in which the i-th element of the [sigma]-th row is a[sigma]_i(i
= 1, ... n, [sigma] = 1, ... k). Denoting the left side of the
[sigma]-th equation by P[sigma]f, it is clear that every common
solution of the two equations P_[sigma]f = 0, P_[rho]f = 0, is also a
solution of the equation P_[rho](P_[sigma]f), P_[sigma](P_[rho]f), We
immediately find, however, that this is also a linear equation,
namely, [Sigma]H_i df/dx_i = 0 where H_i = P[rho]a[sigma]_i -
P[sigma]a[rho]_i, and if it be not already contained among the given
equations, or be linearly deducible from them, it may be added to
them, as not introducing any additional limitation of the possibility
of their having common solutions. Proceeding thus with every pair of
the original equations, and then with every pair of the possibly
augmented system so obtained, and so on continually, we shall arrive
at a system of equations, linearly independent of each other and
therefore not more than n in number, such that the combination, in the
way described, of every pair of them, leads to an equation which is
linearly deducible from them. If the number of this so-called
_complete system_ is n, the equations give df/dx1 = 0 ... df/dxn = 0,
leading to the nugatory result f = a constant. Suppose, then, the
number of this system to be r < n; suppose, further, that from the
matrix of the coefficients a determinant of r rows and columns not
vanishing identically is that formed by the coefficients of the
differential coefficients of f in regard to x1 ... x_r; also that the
coefficients are all deve