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Solve an ODE problem with the right boundary condition at infinity?

Posted 7 months ago
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I have already broken a PDE problem into an ODE problem with the method of separation of variables. But the right boundary locates at infinity. If the right boundary locates at a specific point, I think I can get an orthonormal solution system. But I don't know how to get the eigenvalues by the right boundary at infinity. Please have a look at this issue. Thanks.

eq10 = p1[x] == \[Lambda]*Dp  (p1^\[Prime]\[Prime])[x] // Simplify // 
  Normal

eq11 = Derivative[1][p1][lh] == \[Alpha]1 p1[lh] // Simplify // Normal

eq12 = Derivative[1][p1][\[Infinity]] == 0 // Simplify // Normal

But I don't know how to get the eigenvalues by the right boundary at infinity.

This is usually done by a so-called Ansatz, e.g. if $\lim_{x->\infty}f(x) = 0$ is the boundary condition, one sets $f(x) = g(x) \exp(-x)$; this can work only if $g$ increases less than exponentially, of course. So it needs some insight into the problem to find the right Ansatz.

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