# Lecture Notes for MATH 365 - Ordinary Differential Equation with Buchanan at Millersville (MILL)

## Notes Information

 Material Type: Class Note Professor: Buchanan Class: MATH 365 - Ordinary Differential Equation Subject: Mathematics University: Millersville University of Pennsylvania Term: 2009 Keywords: Robert BuchananSeries SolutionsProvided ThatSubstitutingContinued...ConvergenceDifferentiateDifferentiationHomogeneousSingular Points

## Sample Document Text

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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