# 6 where p is a parameter it turns out that this

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Unformatted text preview: e content of the following theorem, which is proven in more advanced books on diﬀerential equations: Theorem. If the functions P (x) and Q(x) can be represented by power series ∞ ∞ pn (x − x0 )n , P (x) = qn (x − x0 )n Q(x) = n=0 n=0 with positive radii of convergence R1 and R2 respectively, then any solution y (x) to the linear diﬀerential equation d2 y dy + Q(x)y = 0 + P (x) dx2 dx 9 can be represented by a power series ∞ an (x − x0 )n , y (x) = n=0 whose radius of convergence is ≥ the smallest of R1 and R2 . This theorem is used to justify the solution of many well-known diﬀerential equations by means of the power series method. Example. Hermite’s diﬀerential equation is d2 y dy − 2x + 2py = 0, dx2 dx (1.6) where p is a parameter. It turns out that this equation is very useful for treating the simple harmonic oscillator in quantum mechanics, but for the moment, we can regard it as merely an example of an equation to which the previous theorem applies. Indeed, P (x) = −...
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