On the asymptotics of the solution of a system of linear equations with two small parameters
The following Cauchy problem for a system of two linear ordinary differential equations with two independent small parameters multiplying the derivatives is considered: \[ \varepsilon\frac{dx}{dt}=a(t)x+b(t)y+f(t), \mu\frac{dy}{dt}=c(t)x+d(t)y+g(t)\tag{1} \] \[ x|_{t=0}=x^0, y|_{t=0}=y^0,\tag{2} \] where \(a(t),b(t),c(t),d(t),f(t),\) and \(g(t)\) are sufficiently smooth functions of \(t\in[0,T]\) satisfying the conditions \[ a(t)<0, d(t)<0, D(t)=a(t)d(t)-b(t)c(t)>0\tag{3} \] and \(\varepsilon>0\) and \(\mu>0\) are small parameters. Estimates fot the terms in the asymptotic expansion of the solution are obtained. Recursion formulas for the efficient computation of terms of the inner expansion are given.
scientific article; zbMATH DE number 5533503