 Solution

ID: past-exam-of-the-perturbation-methods-and-applied-complex-variables-course-of-the-university-of-oxford/2010/a/1/i/solution

Put

δ = ϵ

y_{0} + δ y_{1/2} + δ^{2} y_{1} + \dots

y_{0}^{'} - x y_{0} = 0, y_{1/2}^{'} - x y_{1/2} = 0, y_{1}^{'} - x y_{1} = y_{0}^{(3)} .

The first two equations give

y_{0} = A e^{x^{2} /2}

and

y_{1/2} = B e^{x^{2} /2}

. Since

y_{0}^{(3)} = A (x^{3} + 3 x) e^{x^{2} /2}

y_{0} = A e^{x^{2} /2}, y_{1/2} = B e^{x^{2} /2}, y_{1} = e^{x^{2} /2} [C + A (\frac{x ^{4}}{4} + \frac{3 x ^{2}}{2})] .

Here

A, B, C

are matching constants; boundary conditions have not yet been imposed.

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