Linear system of differential equations with multiple turning point (Q354834)

The singularly perturbed system of ordinary differential equations \[ y'=A(x)y+A_1(x)y_1, \quad \epsilon y'_1=(B(x)+\epsilon B_1(x))y_1+\epsilon B_2(x)y, \] with \(y\in \mathbb{R}^p,\) \(y_1\in \mathbb{R}^m\) is studied. Using series expansion, the existence of a formal transformation \((y,y_1)^T=\Phi (x,\epsilon)(u,v)^T\) is proved which reduces the original system to an integrable system of equations.