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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 wellknown 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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 Winter '14
 Equations

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