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The Method of Variation of Parameters - Copy (2)

The Method of Variation of Parameters - Copy (2) - A Alaca...

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Unformatted text preview: A. Alaca MATH 1005 Winter 2010 16 The Method of Variation of Parameters For any equation of the form y ′′ + P ( x ) y ′ + Q ( x ) y = G ( x ) , (1) where P ( x ), Q ( x ) and G ( x ) are continuous functions of x , a particular solution can be obtained by variation of parameters. We look for a particular solution of the nonhomogeneous differential equation (1). Let y 1 ( x ) and y 2 ( x ) be any two linearly independent solutions of the AHDE y ′′ + P ( x ) y ′ + Q ( x ) y = 0 (2) Then, we know that the general solution of equation (2) is y c = c 1 y 1 ( x ) + c 2 y 2 ( x ) (3) We seek a particular solution of the non-homogeneous equation (NHE) in the form y p ( x ) = u 1 ( x ) y 1 ( x ) + u 2 ( x ) y 2 ( x ) (4) u 1 ( x ) and u 2 ( x ) to be determined. Since u 1 and u 2 are arbitrary functions, we can impose two conditions on them: • y p is a solution of the diff. eqn. (1) • u ′ 1 y 1 + u ′ 2 y 2 = 0 (to simplify the calculation)....
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