5 The annihilator is a di64256erential operator which when operated on its argument obliterates it There is nothing left Note that there are many functions which cannot be annihilated and hence a di64256erent method must be used to solve th ID: 35779 Download Pdf

5 The annihilator is a di64256erential operator which when operated on its argument obliterates it There is nothing left Note that there are many functions which cannot be annihilated and hence a di64256erent method must be used to solve th

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Math 334 : The Annihilator Section 4.5 The annihilator is a diﬀerential operator which, when operated on its argument, obliterates it. There is nothing left. Note that there are many functions which cannot be annihilated, and hence, a diﬀerent method must be used to solve them. One example is . No linear diﬀerential operator with constant coeﬃcients annihilates it. Here is a short synopsis of all the linear operators which are annihilators and the functions which they annihilate. Annihilator Annihilatee ... αx xe αx αx αx ...

αx αx cos βx αx sin βx αx cos βx αx sin βx xe αx cos βx xe αx sin βx αx cos βx αx sin βx αx cos βx αx sin βx ... ... αx cos βx αx sin βx Note: ( αx Note also that other fuctions can be annihilated besides these. However, they are only known by relating them to the above functions through identities. For example, sinh ) = Annihilator is ( 1)( + 1) = Powers of cos and sin can be annihilated through power reducing formulas and product formulas. Since sin cos 2 and sin

cos [sin( ) + sin( )] then sin = sin cos [sin sin cos 2 ] = sin sin cos 2 sin sin 3 sin( sin sin 3 sin sin sin 3 Thus, it follows that the annihilator of sin is [ + 1][ + 9] since + 1 annihilates sin and + 9 annihilates sin 3 . Note also that this implies that sin ax cos bx can be annihilated as well.

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