ReductionOrder

# ReductionOrder - Reduction of Order Theorem: Let f(x) be a...

This preview shows pages 1–2. Sign up to view the full content.

Reduction of Order Theorem : Let f(x) be a non-trivial solution of the n-th order H.L.D.E. a 0 (x)y n ( ) + a 1 (x)y n ! 1 ( ) + ..... + a n ! 1 (x) " y + a n (x)y = 0 (1) Conclusion : The transformation y = f(x) v(x) reduces the equation (1) to a (n-1)st-order H.L.D.E. in the dependent variable w = dv dx . Let’s consider the case n = 2. We have the 2 nd order equation a 0 (x)y’’ + a 1 (x) y’ + a 2 (x)y = 0 Assume f(x) is a solution of the equation. Take the transformation y = f(x) v(x) where v is a function of x that will be determine at the end of the process. Then dy dx = f(x) dv dx + ! f (x)v d 2 y dx 2 = f(x) d 2 v dx 2 + 2 ! f (x) dv dx + !! f (x)v(x) Substituting in the equation a 0 (x) f(x) d 2 v dx 2 + 2 ! f (x) dv dx + !! f (x)v(x) " # \$ % + a 1 (x) f(x) dv dx + ! f (x)v(x) " # \$ % + a 2 (x)f(x)v(x) = 0 or a 0 (x)f(x) d 2 v dx 2 + 2a 0 (x) ! f (x) + a 1 (x)f(x) [ ] dv dx + a 0 (x) !! f (x)

This preview has intentionally blurred sections. Sign up to view the full version.

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

## This note was uploaded on 07/23/2011 for the course MAP 2302 taught by Professor Staff during the Fall '08 term at FIU.

### Page1 / 4

ReductionOrder - Reduction of Order Theorem: Let f(x) be a...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online