Second order linear ODE with mixed boundary condition - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T02:42:19Z http://mathoverflow.net/feeds/question/64158 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/64158/second-order-linear-ode-with-mixed-boundary-condition Second order linear ODE with mixed boundary condition ProbLe 2011-05-06T21:04:45Z 2011-05-07T18:26:55Z <p>Consider the following second order linear ODE with mixed boundary condition: $$\frac{d^2f}{dt^2}+a(t)f(t)=0,~\frac{df}{dt}(0)=u,~f(1)=0,$$ where $u\in R$ and $a\in C[0,1]$ are fixed.</p> <p>Is the solution to this equation unique? If so, how to prove it? Thanks!</p> http://mathoverflow.net/questions/64158/second-order-linear-ode-with-mixed-boundary-condition/64222#64222 Answer by Deane Yang for Second order linear ODE with mixed boundary condition Deane Yang 2011-05-07T18:26:55Z 2011-05-07T18:26:55Z <p>I'm not sure whether this is really appropriate for MathOverflow or not. Still, let me say a little more: As I've mentioned above, you can work out completely the case where $a$ is constant using explicit solutions. If $a$ is not constant, I'm not aware of a definitive answer but you can get separate necessary conditions and sufficient conditions, involving upper or lower bounds on $a$ using the Sturm comparison theorem. Last, I believe that it is possible to find an integral condition on $a$ that is sufficient for there to be a unique solution to the boundary value problem.</p> <p>The vector-valued version of this problem is used to analyze how geodesics on a Riemannian manifold behave given assumptions on the curvature.</p>