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

Questions
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1

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
2

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.