Past exam of the differential equations 2 course of the University of Oxford / 2014 / 3 / b / i

Seek an order-one outer expansion as $ϵ\to 0^{+}$:At leading order, $0=y_0-x^2$, so $y_0=x^2$. At order $ϵ$,Since $y_0^{\prime \prime }=2$, this gives $y_1=-x^4$. ThereforeThis is an approximation away from any boundary layer; the small coefficient of the highest derivative means the reduced equation cannot generally impose both endpoint conditions.