Reduction of order - Wikipedia, the free encyclopedia

Reduction of order - Wikipedia, the free encyclopedia -...

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Reduction of order - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Reduction_of_order 1 of 3 4/2/08 6:40 AM Reduction of order From Wikipedia, the free encyclopedia Reduction of order is a technique in mathematics for solving second-order ordinary differential equations. It is employed when one solution y 1 ( x ) is known and a second linearly independent solution y 2 ( x ) is desired. A Simple Example Consider the general second-order constant coefficient ODE where a , b , c are real non-zero coefficients. Furthermore, assume that the associated characteristic equation has repeated roots (i.e. the discriminant, b 2 - 4 ac , vanishes). Thus we have Thus our one solution to the ODE is To find a second solution we take as an ansatz where v ( x ) is an unknown function to be determined. Since y 2 ( x ) must satisfy the original ODE, we substitute it back in to get Rearranging this equation in terms of the derivatives of v ( x ) we get Since we know that y 1 ( x ) is a solution to the original problem, the coefficient of the last term is equal to zero. Furthermore,
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