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Unformatted text preview: uation of the form
∞ an xn y (x) =
n=0 and solve for the coeﬃcients an —is surprisingly eﬀective, particularly for the
class of equations called second-order linear diﬀerential equations.
It is proven in books on diﬀerential equations that if P (x) and Q(x) are wellbehaved functions, then the solutions to the “homogeneous linear diﬀerential
+ Q(x)y = 0
+ P (x)
can be organized into a two-parameter family
y = a0 y0 (x) + a1 y1 (x),
called the general solution . Here y0 (x) and y1 (x) are any two nonzero solutions,
neither of which is a constant multiple of the other. In the terminology used in
linear algebra, we say that they are linearly independent solutions. As a0 and
a1 range over all constants, y ranges throughout a “linear space” of solutions.
We say that y0 (x) and y1 (x) form a basis for the space of solutions.
In the special case where the functions P (x) and Q(x) are real analytic,
the solutions y0 (x) and y1 (x) will also be real analytic. This is th...
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This document was uploaded on 01/12/2014.
- Winter '14