285_pdfsam_math 54 differential equation solutions odd

285_pdfsam_math 54 differential equation solutions odd -...

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Exercises 5.2 Thus we have ( AB )[ y ] := A B [ y ] = 2 j =0 a j D j 2 i =0 b i D i [ y ] := 2 j =0 a j D j 2 i =0 b i D i [ y ] := 2 j =0 a j D j 2 i =0 b i D i [ y ] = 2 j =0 2 i =0 ( a j D j b i D i ) [ y ] = 2 i =0 2 j =0 ( b i D i a j D j ) [ y ] = 2 i =0 b i D i 2 j =0 a j D j [ y ] =: 2 i =0 b i D i 2 j =0 a j D j [ y ] =: 2 i =0 b i D i 2 j =0 a j D j [ y ] = B A [ y ] =: ( BA )[ y ] . (b) We have { ( A + B ) + C } [ y ] := ( A + B )[ y ] + C [ y ] := ( A [ y ] + B [ y ]) + C [ y ] = A [ y ] + ( B [ y ] + C [ y ]) =: A [ y ] + ( B + C )[ y ] =: { A + ( B + C ) } [ y ] and { ( AB ) C } [ y ] := ( AB ) C [ y ] := A B C [ y ] =: A ( BC )[ y ] =: { A ( BC ) } [ y ] . (c) Using the linearity of differential operators, we obtain { A ( B + C ) } [ y ] := A ( B + C )[ y ] := A B [ y ] + C [ y ] = A B [ y ] + A C [ y ] =: ( AB )[ y ] + ( AC )[ y ] =: { ( AB ) + ( AC ) } [ y ] . 41. As it was noticed in Example 2, we can treat a “polynomial” in D , that is, an expression of the form p ( D ) = n i =0 a i D i , as a regular polynomial, i.e.,
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