Solving y'' - xy = 0

Differential Equations · Axiom Academy

Use power series to find two independent solutions of the Airy equation about the ordinary point x=0 Find a power series solution to the Airy equation y'' - xy = 0 about the ordinary point x = 0 . Nice work! You solved the Airy equation with the power series method — here's what carries forward to any ordinary-point problem. Series substitution: assume and differentiate term-by-term to get . Index shifting: the xy term produces x^ n+1 , one power ahead of y'' 's x^n — shifting m=n-1 realigns it to so the two sums can combine. Recurrence relation: matching coefficients gives a_2=0 and for — note it links a_ n+2 to a_ n-1 , three indices apart, not two. Two independent solutions: because the recurrence steps by 3, the free constants a_0 and a_1 generate two separate coefficient chains — a_0 drives indices and a_1 drives indices — while every index stays zero. General solution: , where y_1 and y_2 are (up to normalization) the Airy functions and . Because y''-xy=0 has no singular points anywhere in , both series converge for all x — the Airy functions show up in optics, quantum mechanics, and wave theory, describing behavior near a turning point where a wave transitions from oscillatory to exponential.

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