  A new procedure is constructed by means of APS in APLAN language. The
procedure solves the initial-value problem for linear differential equations of
order $k$ with polynomial coefficients and regular singularity in the
initialization point in the interval $[a, b]$ and computes the algebraic
polynomial $y_n$ of given order $n$. A new algorithm of Lanczos $\tau$-method
is built for this procedure, the solution existence $y_n$ of the initial-value
problem proved on this algorithm and also is proved the optimality by precision
of order $k$ derivative of the initial-value problem solution.
