lecture_23 - MA 36600 LECTURE NOTES: WEDNESDAY, MARCH 11...

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: MA 36600 LECTURE NOTES: WEDNESDAY, MARCH 11 Higher Order Linear Equations Review. Recall that a second order linear equation is in the form a ( t ) y 00 + b ( t ) y + c ( t ) y = f ( t ) . We found that the general solution is in the form y ( t ) = c 1 y 1 ( t ) + c 2 y 2 ( t ) + Y ( t ) where { y 1 ( t ) , y 2 ( t ) } is a fundamental set of solutions to the homogeneous equation a ( t ) y 00 + b ( t ) y + c ( t ) y = 0 and Y = Y ( t ) is a particular solution to the nonhomogeneous equation a ( t ) Y 00 + b ( t ) Y + c ( t ) Y = f ( t ) . We give a different way to view this. Define the following operator: L [ y ] = a ( t ) d 2 y dt 2 + b ( t ) dy dt + c ( t ) y. This is a linear operator , i.e., given functions f = f ( t ) and g = g ( t ) as well as constants c 1 and c 2 , we have L c 1 f + c 2 y = a ( t ) d 2 dt 2 c 1 f ( t ) + c 2 g ( t ) + b ( t ) d dt c 1 f ( t ) + c 2 g ( t ) + c ( t ) c 1 f ( t ) + c 2 g ( t ) = a ( t ) c 1 d 2 f dt 2 + c 2 d 2 g dt 2 + b ( t ) c 1 df dt + c 2 dg dt + c ( t ) c 1 f + c 2 g = c 1 a ( t ) d 2 f dt 2 + b ( t ) df dt + c ( t ) f + c 2 a ( t ) d 2 g dt 2 + b ( t ) dg dt + c ( t ) g = c 1 L [ f ] + c 2 L [ g ] . We wish to find all functions y = y ( t ) such that L [ y ] = f ( t ). In a sense, we would like to compute y ( t ) = L- 1 [ f ]. We do this in two steps: First we find functions y 1 = y 1 ( t ) and y 2 = y 2 ( t ) such that L [ y 1 ] = L [ y 2 ] = 0 and W ( y 1 ,y 2 ) ( t ) = y 1 ( t ) y 2 ( t ) y 1 ( t ) y 2 ( t ) = y 1 ( t ) y 2 ( t )- y 1 ( t ) y 2 ( t )...
View Full Document

This note was uploaded on 05/24/2011 for the course MA 36600 taught by Professor Ma during the Spring '09 term at Purdue University-West Lafayette.

Page1 / 3

lecture_23 - MA 36600 LECTURE NOTES: WEDNESDAY, MARCH 11...

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