Find a series expansion, up to xº, for the general solution of Hermite's equation: y" — xy + sy = 0. Write the expansions for the specific solutions y₁, 32, satisfying y₁ (0) = 1, y₁ (0) = 0 and y2 (0) = 0, ; y/₂(0) = 1. Use the recurrence relation to show that the series for either y₁ or for y2 truncates to a polynomial if s is a positive integer. Write down the explicit forms of the expansions for y₁ and y2 for each of the cases s = 3 and s = 4.

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