Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2001 / Paper 2 / 4 / Solution

At an ordinary point, an order-$n$ linear ordinary differential equation has $n$ arbitrary constants in its general solution. Determining them uniquely requires $n$ independent, consistent scalar conditions. Prescribing $y,y^{\prime },\dots ,y^{(n-1)}$ at one ordinary point gives uniqueness under the standard continuous-coefficient assumptions. Conditions at different endpoints need an additional nonsingularity check: for example, $y^{\prime \prime }+y=0$, $y(0)=y(\pi )=0$ leaves every multiple of $\sin x$ possible. Thus counting boundary conditions alone does not guarantee uniqueness for a boundary value problem.