lec_8 - September 21, 2010 8-1 8. Reduction of Order and...

Info iconThis preview shows pages 1–4. 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

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: September 21, 2010 8-1 8. Reduction of Order and more on complex roots Reduction of Order: Suppose we are given a general homogeneous second order d.e. L ( y ) = y 00 + p ( t ) y + q ( t ) y = 0 . (1) We know that, in order to find the general solution, it suffices to find two linearly independent solutions. It turns out that, if we can find one non-zero solution, then a second independent solution can always be found as usual up to integration), by a method called reduction of order. Here is how it works, Suppose y 1 is one non-zero solution to (1). Let us try to find a second solution y 2 = y 1 v where v is a non-constant function. For y 2 to be a solution, we have ( y 1 v ) 00 + p ( y 1 v ) + qy 1 v = 0 or y 00 1 v + 2 y 1 v + y 1 v 00 + py 1 v + py 1 v + qy 1 v = 0 September 21, 2010 8-2 v ( y 00 1 + py 1 + qy 1 ) + v (2 y 1 + py 1 ) + y 1 v 00 = 0 . v (2 y 1 + py 1 ) + y 1 v 00 = 0 (2) since L ( y 1 ) = 0. Now, y 1 and p are known, so we get a first order linear d.e. for v . We solve this for v , then integrate to get v , and then go back to get an actual solution y 2 = y 1 v of L ( y ) = 0. . Since v is not constant, we clearly get that y 2 = y 1 v and y 1 are linearly independent functions. Let us be more specific here. Equation (2) becomes v 00 v =- 2 y 1- p y 1 y 1 (3) d log ( v ) =- 2 y 1- p y 1 y 1 d log ( v ) =- 2 y 1 y 1- p log ( v ) = Z - 2 y 1 y 1- p dt September 21, 2010 8-3 v = exp ( Z - 2 y 1 y 1- p dt ) v = Z exp ( Z - 2 y 1 y 1- p ) dt dt (4) Example 1: The function y ( t ) = t 3 + t is a solution to the d.e....
View Full Document

This note was uploaded on 02/03/2012 for the course MTH 235 taught by Professor Staff during the Fall '11 term at Michigan State University.

Page1 / 11

lec_8 - September 21, 2010 8-1 8. Reduction of Order and...

This preview shows document pages 1 - 4. Sign up to view the full document.

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