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# practice - MA366 Sathaye Solving for yp 1 Practice for New...

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MA366 Sathaye Solving for y p 1. Practice for New Formulas Here f ( D ) is any polynomial in D with constant coefficients. Exponential Slide. f ( D ) e at v = e at f ( D + a ) v. General Product Rule of Leibnitz D n ( vw ) = vD n ( w ) + ( n 1 ) D ( v ) D n - 1 ( w ) + ( n 2 ) D 2 ( v ) D n - 2 ( w ) + · · · + ( n r ) D r ( v ) D n - r ( w ) + · · · + ( n n - 1 ) D n - 1 vD ( w ) + D n ( v ) w = r = n r =0 ( n r ) D r ( v ) D n - r ( w ) As a corollary: f ( D )( vw ) = vf ( D )( w ) + D ( v ) f 0 ( D )( w ) + D 2 ( v ) f 00 ( D )( w ) / 2! + D 3 ( v ) f 000 ( D )( w ) / 3! + · · · . Special cases. f ( D )( tw ) = tf ( D )( w ) + 1 · f 0 ( D ) w Since D 2 ( t ) = 0. More generally f ( D )( t r w ) = t r f ( D )( w ) + rt r - 1 f 0 ( D )( w ) + r ( r - 1) / 2 t r - 2 f 00 ( D )( w ) + · · · 2. Practice on the formulas. (a) Exponential slide. Calculate ( D 2 + 2 D + 1)( e 2 t )( t 2 ). Answer: Let f ( D ) = D 2 + 2 D + 1 = ( D + 1) 2 . Then f ( D )( e 2 t t 2 ) = e 2 t ( f ( D +2)( t 2 )) = e 2 t (( D +2)+1) 2 ( t 2 ) = e 2 t ( D +3) 2 ( t 2 ) = e 2 t (2+6(2 t )+9 t 2 ) . Calculate ( D 2 + 2 D + 1)( e - t )( t 2 ). Answer: Let f ( D ) = D 2 + 2 D + 1 = ( D + 1) 2 . Then f ( D )( e - t t 2 ) = e - t ( f ( D - 1)( t 2 )) = e - t (( D - 1) + 1) 2 ( t 2 ) = e - t D 2 ( t 2 ) = e - t (2) . Calculate ( D 3 + 3 D + 4) e 5 t . Answer: ( D 3 + 3 D + 4)( e 5 t (1)) = e 5 t (( D + 5) 3 + 3( D + 5) + 4)(1) = e 5 t (5 3 + 3(5) + 4) . The last calculation is based on the fact that since D ( c ) = 0 for any constant c , we have D r ( c ) = 0 when r > 0 and thus g ( D )( c ) = g (0) c . 1

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(b) Reduction of order. The Leibnitz’ rule for polynomial operators gives a fast way to carry out the reduction of order calculations. Let ( t 2 D 2 + 2 tD - 2) y = 0 have one given equation y 1 = t . Find the other by reduction of order. Answer: Let f ( D ) = t 2 D 2 + 2 tD - 2. Then f 0 ( D ) = 2 t 2 D + 2 t and f 00 ( D ) = 2 t 2 . Note that f 000 ( D ) = 0. Put y = tw .
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practice - MA366 Sathaye Solving for yp 1 Practice for New...

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