ReductionOrder

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

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

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

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