3 - init_cond - Series sol. near ordinary points w/ given...

Info iconThis preview shows pages 1–3. 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: Series sol. near ordinary points w/ given initial conds. Lets revisit the 2 problems that we have solved in class but lets solve them with some initial conditions. We will solve them two using two different methods. 1. Using the results we obtained in ordinary_ps.pdf . 2. Using a Taylor series expansion of the solution about x . Before doing so, recall that the Talor series expansion of a function about point x is given by y ( x ) = y ( x ) + y ( x )( x- x ) + y 00 ( x ) 2! ( x- x ) 2 + y 000 ( x ) 3! ( x- x ) 3 + y (4) ( x ) 4! ( x- x ) 4 + . (1) For a given second order initial value problem (IVP), we already know y ( x ) and y ( x ) . What we need to find are y 00 ( x ) ,y 000 ( x ) , . We will find these using the given differential equation, as illustrated in the problems below. 1 c HF, 2011 Series sol. near ordinary points w/ given initial conds. Problem 1 (1 + x ) y 00 + y = 0 , x = 0 . y (0) = 1 ,y (0) = 1 . (2) Solution Method A : Recall that we previous found the general solution to the problem to be (see ordinary_ps.pdf ) y = a 1- 1 2 x 2 + 1 6 x 3- 1 24 x 4 + + a 1 x- 1 6 x 3 + 1 12 x 4 + , (3) where a and a 1 are two arbitrary constants. Using the first initial conditions, we find y (0) = 1 = a 1- 1 2 2 + 1 6 3- 1 24 4 + + a 1- 1 6 3 + 1 12 4 + a = 1 ....
View Full Document

Page1 / 5

3 - init_cond - Series sol. near ordinary points w/ given...

This preview shows document pages 1 - 3. Sign up to view the full document.

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