Partial Differentials Notes_Part_8

Partial Differentials Notes_Part_8 - Remark : The...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Remark : The definition of a singular point assumes that the other coefficients do not both vanish there, i.e., either q ( x ) negationslash = 0 or r ( x ) negationslash = 0. If all three functions happen to vanish at x , we can cancel any common factor ( x x ) k , and hence, without loss of generality, can assume at least one of the coefficients is nonzero at x . Proofs of the basic existence theorem for differential equations at regular points can be found in [ 66 , 72 ]. Theorem 12.5. Let x be a regular point for the second order homogeneous linear ordinary differential equation (12.69) . Then there exists a unique, analytic solution u ( x ) to the initial value problem u ( x ) = a, u ( x ) = b. (12 . 71) The radius of convergence of the power series for u ( x ) is at least as large as the distance from the regular point x to the nearest singular point of the differential equation in the complex plane. Thus, every solution to an analytic differential equation at a regular point x can be expanded in an convergent power series u ( x ) = u + u 1 ( x x ) + u 2 ( x x ) 2 + = summationdisplay n = 0 u n ( x x ) n . (12 . 72) Since the power series necessarily coincides with the Taylor series for u ( x ), its coefficients u n = u ( n ) ( x ) n ! are multiples of the derivatives of the function at the point x . In particular, the first two coefficients u = u ( x ) = a, u 1 = u ( x ) = b. (12 . 73) are fixed by the initial conditions. Once the initial conditions have been specified, the remaining coefficients must be uniquely prescribed thanks to the uniqueness of solutions...
View Full Document

This note was uploaded on 02/10/2012 for the course MATH 5587 taught by Professor Olver during the Fall '10 term at University of Central Florida.

Page1 / 3

Partial Differentials Notes_Part_8 - Remark : The...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online