As a0 and a1 range over all constants y ranges

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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 equation” d2 y dy + Q(x)y = 0 + P (x) dx2 dx 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.

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