4.4we - 4.4: THE METHOD OF VARIATION OF PARAMETERS KIAM...

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4.4: THE METHOD OF VARIATION OF PARAMETERS KIAM HEONG KWA The goal of this section is to generalize the method of variation of parameters for second-order linear equations in section 3.7 to the class of higher order linear equations. For concreteness, we consider the equation (1) y ( n ) + p 1 ( t ) y ( n - 1) + ··· + p n - 1 ( t ) y 0 + p n ( t ) y = g ( t ) , where p i ( t ), i = 1 , 2 , ··· ,n , and g ( t ) are continuous functions in an open interval I . As we have indicated in section 4.1, if { y i ( t ) | i = 1 , 2 , ··· ,n } is a fundamental set of solutions of the corresponding ho- mogeneous equation, then the general solution of the homogeneous equation is given by (2) y h ( t ) = n X i =1 c i y i ( t ) , where c i , i = 1 , 2 , ··· ,n , are integration constants. The idea of the method of variation of parameters is to assume a priori that the general solution y ( t ) of (1) based on (2) by substituting for the constants c i with some differentiable functions u i ( t ) to be determined: (3) y ( t ) = n X i =1 u i ( t ) y i ( t ) . By differentiating (3), we obtain y 0 ( t ) = n X i =1 u i ( t ) y 0 i ( t ) + n X i =1 u 0 i ( t ) y i ( t ) . To avoid leading to higher order derivatives of
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4.4we - 4.4: THE METHOD OF VARIATION OF PARAMETERS KIAM...

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