notes 5-3

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

This preview shows pages 1–2. Sign up to view the full content.

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!!

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online