Godric's solution is a good one, but sometimes there's a better way of doing the shooting method for asymptotic boundary conditions.

Basically the idea is to approximate the ODE with another ODE which captures the behavior of the original in the large $x$ limit. For example, a term like  $$\left(x-\alpha\right)\frac{d \phi}{dx}$$ would become $$x\frac{d \phi}{dx}$$

If the resulting differential equation is exactly solvable, you can pick the exact solution that has the asymptotic behavior you require and some $x$ large enough that the asymptotic approximation is good, and then use the shooting method to find the matching point.

Something like this technique also works for oscillatory asymptotic boundary conditions.

If you find yourself doing these kinds of problems a lot, you might want to check out Bender and Orszag's eldrich tome of asymptotics lore.

Godric's solution is a good one, but sometimes there's a better way of doing the shooting method for asymptotic boundary conditions.

Basically the idea is to approximate the ODE with another ODE which captures the behavior of the original in the large $x$ limit. For example, terms like 

$$\left(x-\alpha\right)\frac{d \phi}{dx}$$ or $$\frac{\beta}{x^2}\phi $$

would become $$x\frac{d \phi}{dx}$$ and $$0$$

If the resulting differential equation is exactly solvable, you can pick the exact solution that has the asymptotic behavior you require and some $x$ large enough that the asymptotic approximation is good, and then use the shooting method to find the matching point.

Something like this technique also works for oscillatory asymptotic boundary conditions.

If you find yourself doing these kinds of problems a lot, you might want to check out Bender and Orszag's eldrich tome of asymptotics lore.

Godric's solution is a good one, but sometimes there's a better way of doing the shooting method for asymptotic boundary conditions.

Basically the idea is to approximate the ODE with another ODE which captures the behavior of the original in the large $x$ limit. For example, a term like $$\left(x-\alpha\right)\frac{d \phi}{dx}$$ would become $$x\frac{d \phi}{dx}$$

If the resulting differential equation is exactly solvable, you can pick the exact solution that has the asymptotic behavior you require and some $x$ large enough that the asymptotic approximation is good, and then use the shooting method to find the matching point.

This also works for oscillatory asymptotic boundary conditions.

Godric's solution is a good one, but sometimes there's a better way of doing the shooting method for asymptotic boundary conditions.

Basically the idea is to approximate the ODE with another ODE which captures the behavior of the original in the large $x$ limit. For example, a term like $$\left(x-\alpha\right)\frac{d \phi}{dx}$$ would become $$x\frac{d \phi}{dx}$$

If the resulting differential equation is exactly solvable, you can pick the exact solution that has the asymptotic behavior you require and some $x$ large enough that the asymptotic approximation is good, and then use the shooting method to find the matching point.

Something like this technique also works for oscillatory asymptotic boundary conditions.

If you find yourself doing these kinds of problems a lot, you might want to check out Bender and Orszag's eldrich tome of asymptotics lore.