Power asymptotics of solutions of a system of differential equations, not solved for the derivative (Q1100335)

We consider a system of the form \[ (1)\quad t^ A(dX/dt)=BX+\Phi (t)+X^ 2F(t,X,dX/dt), \] where \(t\in (0,t_ 0)\), \(A=diag(a_ 1,...,a_ n)\) and \(B=diag(b_ 1,...,b_ n)\) are constant matrices, all \(a_ i>1\), \(X=diag(x_ 1,...,x_ n)\), \(\Phi =diag(\phi_ 1,...,\phi_ n)\), \(F=diag(f_ 1,...,f_ n)\) is a matrix function, continuous and having continuous partial derivative in dX/dt in a neighborhood of the zero values of its variables. We study the question of existence and asymptotics of solutions of (1), having the property: \(X\to 0\) as \(t\to +0\).