4-8. Variation of parameters(Homogeneous linear equations with constant coefficients)

4-8. Variation of parameters(Homogeneous linear equations with constant coefficients)

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Variation of parameters Let given a linear second-order differential equation a 2 ( x ) y ' ' + a 1 ( x ) y ' + a 0 ( x ) y = g ( x )( 1 ) For the linear second-order differential equation (1) we seek a particular solution in the form y p = u 1 ( x ) y 1 + u 2 ( x ) y 2 where y 1 and y 2 form a fundamental set of solutions on I for a 2 ( x ) y ' ' + a 1 ( x ) y ' + a 0 ( x ) y = 0 of the associated homogeneous form (1). It can be checked easily that if function u 1 ( x ) , ¿ u 2 ( x ) ¿ is solution of the system { y 1 u 1 ' + y 2 u 2 ' = 0 y 1 ' u 1 ' + y 2 ' u 2 ' = f ( x ) ( 2 ) then y p = u 1 ( x ) y 1 + u 2 ( x ) y 2 is a particular solution for (1). By the Cramer’s Rule, the solution of system (2) can be expressed in the terms of determinants: u 1 ' = W 1 W = y 2 f ( x ) W , u 2 ' = W 2 W = y 1 f ( x ) W ( 3 ) where W = | y 1 y 2 y 1 ' y 2 ' | W 1 = | 0 y 2 f ( x ) y 2 ' | ,W 2 = | y 1 0 y 1 ' f ( x ) | (4)
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The function u 1 ( x ) and u 2 ( x ) are found by integrating (3). The determinant W is called Wronskian of y 1 and y 2 . By linear independence of y 1 and y 2 we know that W ≠ 0. Example 1. Solve y '' 4 y ' + 4 y =( x + 1 ) e x .
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