 Solution

ID: past-exam-of-the-perturbation-methods-and-applied-complex-variables-course-of-the-university-of-oxford/2009/a/3/ii/solution

ϵ y^{''}

against

x y

x = γ X

, their equality requires

ϵ / γ^{2} = γ

, hence

γ = ϵ^{1/3}

. The rescaled equation is

Y^{''} + ϵ^{1/3} (ϵ^{1/3} X - 1) Y^{'} - X Y = 0.

Y^{''} = X Y

. Thus

Y = α Ai (X) + β Bi (X) .

Let

a_{0} = Ai (0) = 3^{- 2/3} /Γ (2/3)

; then

Bi (0) = 3 a_{0}

. The left boundary condition becomes

a_{0} (α + 3 β) = 1

, or

β = 3^{1/6} Γ (2/3) - \frac{α}{3} .

This determines one linear relation before the far-boundary matching condition is used.

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