Abstract: This paper deals with the singular perturbation of the boundary value problem of the systems for quasi-linear ordinary differential equations x'=f(t,x,y,ε),x(0,ε)=A(ε) εy"=g(t,x,y,ε)y'+h(t,x,y,ε) y(0,ε)=B(ε),y(1,ε)=C(ε) where x,f, y, h, A, B and C all belong to Rⁿ, and g is an n×n matrix function. Under suitable conditions we prove the existence of the solutions by diagonalization and the fixed point theorem and also estimate the remainder.

Key words: systems of the quasi-linear ordinary differential equation, singular perturbation, diagonalization, asymptotic expansion