initial conditions in desolve

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I believe that only Dirichlet and Neumann boundary conditions are implemented (but would happy to be proven wrong).

The relevant part of the docs is:

• for a second-order equation, specify the initial x, y, and dy/dx, i.e. write [x_0, y(x_0), y'(x_0)]

• for a second-order boundary solution, specify initial and final x and y boundary conditions, i.e. write [x_0, y(x_0), x_1, y(x_1)].

So the first to boundary conditions you gave are ok. But the mixed boundary conditions don't work - you'll have to get the general solution and enforce the boundary conditions semi-manually. Here's a sage notebook with examples.