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380_pdfsam_math 54 differential equation solutions odd

380_pdfsam_math 54 differential equation solutions odd -...

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Chapter 6 To find functions v j ’s we need four determinants, the Wronskian W [ y 1 , y 2 , y 3 ]( x ) and W 1 ( x ), W 2 ( x ), and W 3 ( x ) given in (10) on page 340 of the text. Thus we compute W e x , e 2 x , xe 2 x ( x ) = e x e 2 x xe 2 x e x 2 e 2 x (1 + 2 x ) e 2 x e x 4 e 2 x (4 + 4 x ) e 2 x = e x e 2 x e 2 x 1 1 x 1 2 1 + 2 x 1 4 4 + 4 x = 9 e 3 x , W 1 ( x ) = ( 1) 3 1 W e 2 x , xe 2 x ( x ) = e 2 x xe 2 x 2 e 2 x (1 + 2 x ) e 2 x = e 4 x , W 2 ( x ) = ( 1) 3 2 W e x , xe 2 x ( x ) = e x xe 2 x e x (1 + 2 x ) e 2 x = (1 + 3 x ) e x , W 3 ( x ) = ( 1) 3 3 W e x , e 2 x ( x ) = e x e 2 x e x 2 e 2 x = 3 e x . Substituting these expressions into the formula (11) for determining v j ’s, we obtain v 1 ( x ) = g ( x ) W 1 ( x ) W [ e x , e 2 x , xe 2 x ] dx = e 2 x e 4 x 9 e 3 x dx = 1 27 e 3 x , v 2 ( x ) = g ( x ) W 2 ( x ) W [ e x , e 2 x , xe 2 x ] dx = e 2 x (1 + 3 x ) e x 9 e 3 x dx = 1 9 (1 + 3 x ) dx = x 9 x 2 6 , v 3 ( x ) = g ( x ) W 3 ( x ) W [ e x , e 2 x , xe 2 x ] dx = e 2 x 3 e x 9 e 3 x dx = x 3 , where we have chosen zero integration constants. Then formula (12), page 340 of the text, gives a particular solution y p ( x ) = 1 27 e 3 x e x x 9 + x
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