notes 5-3 - SECTION 5.3 SOME THEORY FOR POWER SERIES...

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SECTION 5.3 SOME THEORY FOR POWER SERIES SOLUTIONS When P ( x ) ,Q ( x ) and R ( x ) are polynomials – or even just when they can be represented by power series – and P ( x 0 ) 6 = 0 (so that a solution is guaranteed), we can plug φ ( x ) = X n =0 a n ( x - x 0 ) n into the differential equation P ( x ) y 00 + Q ( x ) y 0 + R ( x ) y = 0 and determine the values of the a n using a recurrence relation. Then to approximate values of y ( x ) we can just add up a bunch of terms of the series. But how do we know when to stop adding, and when we do, how do we know how close we are to the exact value of y ( x )? For example, we could stop adding when the next term in the series is, say, less than half of some acceptable error. EXAMPLE. Suppose we do all this and we find that φ ( x ) = X n =0 n n 2 + 1 x n . Our problem needs the value φ (1) with an error of no more than 0.01. How many terms must we add up? WRONG!!
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notes 5-3 - SECTION 5.3 SOME THEORY FOR POWER SERIES...

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