MATH-S 343 Chapter 5: S343 5.2 Notes (Nov. 8)
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S343 section 5. 2 notes- series solutions near an ordinary point (part 1) 3. 2. 1 does not apply procedure for solving non-homogeneous similar (cid:4666)(cid:1876) (cid:1876)(cid:2868)(cid:4667) By existence/uniqueness theorem 3. 2. 1, there exists in that interval a unique solution satisfying initial: differentiation justified as long as within interval of convergence. Seems unattractive, but form is convenient- power series behave like polynomials, easy to manipulate algebraically. Solutions in terms of elementary functions (ie. exponentials/trig) often need power series representation to be graphed/manipulated easily. Focus on second order equations with coefficients that are functions of independent variable (cid:1876: (cid:1842)(cid:4666)(cid:1876)(cid:4667)(cid:3118)(cid:3052)(cid:3051)(cid:3118)+(cid:1843)(cid:4666)(cid:1876)(cid:4667)(cid:3052)(cid:3051)+(cid:1844)(cid:4666)(cid:1876)(cid:4667)(cid:1877)=(cid:882, primarily consider case in which (cid:1842),(cid:1843),(cid:1844) are polynomials; also applicable to general analytic functions. Suppose no factor (cid:4666)(cid:1876) (cid:4667) common to all 3 of (cid:1842),(cid:1843),(cid:1844); if there is, divide out first. In that interval, divide by (cid:1842)(cid:4666)(cid:1876)(cid:4667) to get (cid:1877) +(cid:1868)(cid:4666)(cid:1876)(cid:4667)(cid:1877) +(cid:1869)(cid:4666)(cid:1876)(cid:4667)(cid:1877)=(cid:882), where (cid:1868)(cid:4666)(cid:1876)(cid:4667)=(cid:3018)(cid:4666)(cid:3051)(cid:4667)(cid:3017)(cid:4666)(cid:3051)(cid:4667),(cid:1869)(cid:4666)(cid:1876)(cid:4667)=(cid:3019)(cid:4666)(cid:3051)(cid:4667)(cid:3017)(cid:4666)(cid:3051)(cid:4667) conditions (cid:1877)(cid:4666)(cid:1876)(cid:2868)(cid:4667)=(cid:1877)(cid:2868),(cid:1877) (cid:4666)(cid:1876)(cid:2868)(cid:4667)=(cid:1877)(cid:2868) for arbitrary values of (cid:1877)(cid:2868),(cid:1877)(cid:2868) : singular point- (cid:1876)(cid:2868) such that (cid:1842)(cid:4666)(cid:1876)(cid:2868)(cid:4667)=(cid:882)
