If I have a partial differentail equation such as $\frac{\partial^2 u}{\partial t^2} + c^2 \frac{\partial^2 u}{\partial x^2}$, with boundary conditions $u(0,t) = 0$ and $u(1,t)=0$, I can solve this system using separation of variables: $u=X(x)T(t)$. This means I have an ODE: $X''(x) + (c^2-\lambda)X(x)=0$, with $\lambda$ as separation constant.

To look for solutions I set up the Wronskian and search for $\lambda$ values for which the Wronskian determinant is zero, meaning I found an eigenvalue. For 'easy'boundary conditions this can also be done manually, however I want to do it numerically.

Now I look over a range of $\lambda$, look for minima, and optimize around those minima.

Knowing how the Wronskian is set up, is there an optimization method or numerical trick I can perform to speed up the process?


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