An algorithm computing the regular formal solutions to a system of linear differential equations
The authors consider the first-order linear differential system \[ x{dY\over dx}= A(x)Y.\tag{1} \] It writes \(A= x^{-q}(A_0+ A_1x+\cdots)\), \(A_0\neq 0\), for the series expansion of \(A\), where the coefficients are matrices over a subfield \(K\) of the field of complex numbers. There exists a basis of \(n\) formal solutions of the form \[ y_i(t)= e^{q_i(t)} t^{\lambda_i} z_i(t),\quad i= 1,\dots, n, \] where \(t^{r_i}= x\) for positive integers \(r_i,q_i\in t^{-1} \overline{K}[t^{-1}]\), \(\lambda_i\in\overline K\) and \(z_i\in\overline Kt^n[\log(t)]\). These solutions form the columns of a formal fundamental matrix solution to (1) which can be written as \[ U(t)= H(t) t^\Lambda e^{Q(t)} \] with \(t^r= x\) for a positive integer \(r, H\in M_n\overline Kt\) is a formal matrix power series, \(\Lambda\in M_n\overline K\) is a constant matrix with eigenvalues \(\lambda_1,\dots, \lambda_n\) and \(Q= \text{diag}(q_1,\dots, q_n)\). The problem is to find a nonzero solution to (1) of the form \[ y(x)= x^\lambda \sum^\infty_{i=0} g_i x^i,\quad g_0\neq 0, \] or decide that there is no such solution. Insight is gained if one considers systems of the more general form, \(D\theta(Y)= NY\), where \(\theta= x{d\over dx}\), \(D\) and \(N\) are formal power series matrices without a pole. Note that the matrix \(D^{-1}\) may have a pole. The main result is then within software MAPLE, an algorithm is developed giving the solutions to be \[ y_s(x)= h_s(x)+ \log(x) h_{s- 1}(x)+\cdots+ {\log^{s- 1}(x)\over(s- 1)!} h_1(x), \] with \(h_i\in \overline K((x))^n\).
scientific article; zbMATH DE number 1440304
scientific article