Lecture_15 - MA 36600 LECTURE NOTES FRIDAY FEBRUARY 20 We explain how this is related to Abels Theorem Using the the characteristic equation are b

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
MA 36600 LECTURE NOTES: FRIDAY, FEBRUARY 20 3 We explain how this is related to Abel’s Theorem. Using the Quadratic Formula, we see that the roots of the characteristic equation are r 1 = b 2 a b 2 4 ac 2 a r 2 = b 2 a + b 2 4 ac 2 a = r 1 + r 2 = b a r 2 r 1 = b 2 4 ac a Hence the Wronskian is in the form W ( t )= C exp ° b a t ± in terms of C = b 2 4 ac a . Since b 2 4 ac 0 by assumption, we see that this function is nonzero if and only if b 2 4 ac> 0. Applications of Abel’s Theorem. First, we show that if W ( t 0 ) ° = 0 for some t 0 then y = c 1 y 1 + c 2 y 2 is the general solution to the di±erential equation. To see why, we can consider some initial conditions y ( t 0 )= y 0 and y ° ( t 0 )= y ° 0 . Since W 0 = W ( t 0 ) ° = 0 we can solve for c 1 and c 2 . Note that W ( t 0 ) ° = 0 precisely when the constant C ° = 0. Second, we give a method for ²nding solutions to the di±erential equation. Say that we know one solution y 1 = y 1 (
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.

Ask a homework question - tutors are online