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the following theorem, which is proven in more advanced books on diﬀerential
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
+ Q(x)y = 0
+ P (x)
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
+ 2py = 0,
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
P (x) = −...
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- Winter '14