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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 Buchanan
  • Series Solutions
  • Provided That
  • Substituting
  • Continued...
  • Convergence
  • Differentiate
  • Differentiation
  • Homogeneous
  • Singular Points
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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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