lec_12b-10-11-10-an

# lec_12b-10-11-10-an - 12b-1 12b Series Solutions of Second...

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Unformatted text preview: August 18, 2010 12b-1 12b. Series Solutions of Second Order Linear Differential Equations near an ordinary point We now apply the notions of power series to second order linear differential equations. Consider such an equation (with independent variable x ) P ( x ) y 00 + Q ( x ) y + R ( x ) y = g ( x ) (1) in which each of the functions P ( x ) ,Q ( x ) ,R ( x ) ,g ( x ) is real analytic in an interval of about the base point a . If P ( a ) 6 = 0, then we call a an ordinary point of (1). Otherwise, we call a a singular point of (1). In this section we investigate solutions near an ordinary point. We have the following Theorem. Theorem. Assume that the functions P,Q,R,g are real analytic in an interval I which contains the point a , and let R > be less than the minimum of the radii of convergence of the functions P,Q,R,g expanded in Taylor series with base point a . Assume that ( a- R,a + R ) ⊂ I and that P ( x ) 6 = 0 for all x ∈ ( a- R,a + R ) . Then, given constants C ,C 1 there is a unique so- lution y ( x ) to the initial value problem August 18, 2010 12b-2 P ( x ) y 00 + Q ( x ) y + R ( x ) y = g ( x ) , y ( a ) = C , y ( a ) = C 1...
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lec_12b-10-11-10-an - 12b-1 12b Series Solutions of Second...

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