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Series Solutions Near an Ordinary Point
MATH 365 Ordinary Differential Equations
J. Robert Buchanan
Department of Mathematics
Summer 2009
J. Robert Buchanan Series Solutions Near an Ordinary Point
Ordinary Points (1 of 2)
Consider the second order linear homogeneous ODE:
P(t)y'' + Q(t)y' + R(t)y = 0
where P, Q, and R are polynomials.
Definition
A point t0 such that P(t0) negationslash= 0 is called an ordinary point. If
P(t0) = 0 then t0 is called a singular point.
J. Robert Buchanan Series Solutions Near an Ordinary Point
Ordinary Points (2 of 2)
P(t)y'' + Q(t)y' + R(t)y = 0
If t0 is an ordinary point, then by continuity there exists an
interval (a, b) containing t0 on which P(t) negationslash= 0 for all t ? (a, b).
Thus the functions p(t) = Q(t)P(t) and q(t) = R(t)P(t) are defined
and continuous on (a, b) and the ODE can be written as
y'' + p(t)y' + q(t)y = 0.
If the initial conditions are y(t0) = y0 and y'(t0) = y'0 then there
exists a unique solution to the ODE satisfying the initial
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