notes2 - y 1 y 2 y 1 y 2 u 1 u 2 = f ( x ) ; with the...

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MATH 1005A - Notes 2 Variation of Parameters Consider a linear, nonhomogeneous equation in standard form, y 00 + p ( x ) y 0 + q ( x ) y = f ( x ) ; and let y 1 and y 2 be two, independent solutions of the associated homogeneous equation y 00 + p ( x ) y 0 + q ( x ) y = 0. We seek a particular solution y p of the nonhomogeneous equation in the form y p ( x )= u 1 ( x ) y 1 ( x )+ u 2 ( x ) y 2 ( x ) ;u 1 and u 2 to be determined. Then y p = u 1 y 1 + u 2 y 2 ) y 0 p = u 0 1 y 1 + u 1 y 0 1 + u 0 2 y 2 + u 2 y 0 2 =( u 0 1 y 1 + u 0 2 y 2 )+( u 1 y 0 1 + u 2 y 0 2 ) : The requirement that y p be a solution of the nonhomogeneous equation imposes one condition upon the functions u 1 and u 2 . Since there are two functions to be determined, we may impose a second condition upon them and, thus, we require that u 0 1 y 1 + u 0 2 y 2 =0 . Then y 0 p = u 1 y 0 1 + u 2 y 0 2 ) y 00 p = u 1 y 00 1 + u 0 1 y 0 1 + u 0 2 y 0 2 + u 2 y 00 2 ; and y p is a solution if and only if [ u 1 y 00 1 + u 0 1 y 0 1 + u 0 2 y 0 2 + u 2 y 00 2 ]+ p ( x )[ u 1 y 0 1 + u 2 y 0 2 ]+ q ( x )[ u 1 y 1 + u 2 y 2 ]= f ( x ) ; i.e., u 1 [ y 00 1 + p ( x ) y 0 1 + q ( x ) y 1 ]+ u 2 [ y 00 2 + p ( x ) y 0 2 + q ( x ) y 2 ]+ u 0 1 y 0 1 + u 0 2 y 0 2 = f ( x ) : Since y 1 and y 2 are solutions of the homogeneous equation y 00 + p ( x ) y 0 + q ( x ) y = 0, the latter requirement reduces to u 0 1 y 0 1 + u 0 2 y 0 2 = f ( x ). Combining with the ¯rst condition imposed, we obtain the system of equations u 0 1 y 1 + u 0 2 y 2 =0 u 0 1 y 0 1 + u 0 2 y 0 2 = f ( x ) for the unknown quantities u 0 1 and u 0 2 . The system may be expressed in matrix form as
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Unformatted text preview: y 1 y 2 y 1 y 2 u 1 u 2 = f ( x ) ; with the solution u 1 u 2 = y 1 y 2 y 1 y 2 1 f ( x ) = 1 y 1 y 2 y 1 y 2 y 2 y 2 y 1 y 1 f ( x ) : The quantity y 1 y 2 y 1 y 2 = y 1 y 2 y 1 y 2 is denoted by W ( x ) = W [ y 1 ( x ) y 2 ( x )] and called the Wronskian of the functions y 1 and y 2 . Thus, 2 u 1 u 2 = 1 W ( x ) y 2 y 2 y 1 y 1 f ( x ) = 1 W ( x ) y 2 f ( x ) y 1 f ( x ) ) u 1 = y 2 f W and u 2 = y 1 f W ) u 1 = Z y 2 f W dx and u 2 = Z y 1 f W dx:...
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notes2 - y 1 y 2 y 1 y 2 u 1 u 2 = f ( x ) ; with the...

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