Fyrilestrar+STAE+III+verkfraedi+-+skammtur+4

# u y1 py1 qy1 v y2 py2 qy2 u y1

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

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: y1 = u￿ (y1 )2 = (y1 )2 e−G(x) ￿= 0, y1 y2 ✷ y1 (x) = x2 3 4 y ￿￿ − y ￿ + 2 y = 0, x x x>0 y2 = uy1 = ux2 ￿ y2 = u￿ x2 + u2x ￿￿ y2 = u￿￿ x2 + u￿ 2x + u￿ 2x + 2u = x2 u￿￿ + 4xu￿ + 2u. 3 4 ! x2 u￿￿ + 4xu￿ + 2u − (x2 u￿ + 2xu) + 2 ux2 = 0, x x ! x2 u￿￿ + xu￿ = 0. v = u￿ x xv ￿ + v = 0 (xv )￿ = 0 v k x v= u￿ = v u￿ = k x u = k ln(x) + c k=1 y2 = uy1 = ln(x)x c=0 u = ln(x) 2 ✷ ay ￿￿ + by ￿ + cy = 0 y ￿￿ + p(x)y ￿ + q (x)y = f (x) y ￿￿ + p(x)y ￿ + q (x)y = 0 y1 y2 y ￿￿ + p(x)y ￿ + q (x)y = 0 (∗). y (x) = c1 · y1 (x) + c2 · y2 (x), c1 , c2 ∈ R. c1 c2 u(x) v (x) yp (x) = u(x)y1 (x) + v (x)y2 (x) y ￿￿ + p(x)y ￿ + q (x)y = f (x) (∗∗). ￿ ￿ ￿ yp = u￿ y1 + uy1 + v ￿ y2 + vy2 . u ￿ y 1 + v ￿ y 2 = 0. ￿ ￿ ￿ yp = uy1 + vy2 ￿￿ ￿ ￿￿ ￿ ￿￿ yp = u￿ y1 + uy1 + v ￿ y2 + vy2 . (∗∗) ! ￿ ￿￿ ￿ ￿￿ ￿ ￿ u￿ y1 + uy1 + v ￿ y2 + vy2 + p · [uy1 + vy2 ] + q · [uy1 + vy2 ] = f. ! ￿￿ ￿ ￿￿ ￿ ￿ ￿ u · [y1 + py1 + qy1 ] + v · [y2 + py2 + qy2 ] + u￿ y1 + v ￿ y2 = f, y1 y2 (∗) ! ￿ ￿ u￿ y1 + v ￿ y2 = f. u￿ u￿ (x) = − v￿ y2 (x) · f (x) W (x)...
View Full Document

## This document was uploaded on 02/16/2014.

Ask a homework question - tutors are online