On a singular perturbation problem with two second-order turning points

In this paper, we study the singular perturbation problem ε p ( x ) u xx + 2 ( x - a ) ( b - x ) q ( x ) u x - 2 su = 0 , 0 < x < 1 , 0 < a < b < 1 , where 0 < ε ⪡ 1 is a small positive parameter, p ( x ) and q ( x ) are sufficiently smooth and strictly positive functions. The main feature of this equation is that there are two second-order turning points in the interval ( 0 , 1 ) . Based on the rigorous results on singular perturbation problems with one second-order turning point in our previous work, we obtain a uniform asymptotic approximation for the general solution of the above equation by means of a matching technique.