please help me with this D-operator problem?

Question

Verify that the given differential operator annihilates the given function
(i) 2D-1; y = 4e^(x/2)
(ii)D^4 ; y=10x^3 -2 x
please help me with some explanation about how you work the above out

Answer 1

y = 4e^(x/2) ======> 2D - 1



4e^(x/2) (2D - 1)



2D 4e^(x/2) - 4e^(x/2)



8 * ( (1/2) e^(x/2) ) - 4e^(x/2)



4e^(x/2) - 4e^(x/2)



0 <===== true



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y = 10x^3 - 2x <====== D^4 means the fourth derivative



y ' 30x^2 - 2



y '' = 60x - 2



y ''' = 60



y '''' = 0



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please e-mail if have a question.

Answer 2

(I). Dy=y/2 because differentiation brings a factor 1/2 into play. hence 2D-1 Applied to y is zero.



(ii). If you differentiate any cubic polynomial 4 Times you will get zero. The highest term in this case, 10x^3, becomes successively 30x^2, 60x, 60, and then zero. All the other terms vanish more quickly.

Answer 3

very well bhai