9. Let f(x) = (1 + 12)-4. (a) Show that (1 + x) f(x) +if(x) = 0. (b) Let n be a non-negative integer, show that (1 + z?) f ( n+2) (x) + (2n + 3)xf(n+1)(x) + (n +1)2f(n)(x) =0. term r. (c) Hence, or otherwise, find the Taylor series of (1 + z?) ? at x = 0 up to the (1 + x" ) flx ) + xtla ) = 0 fix) - Clax) f'(x ) = - * (1x ) dX7 ( 1 +X" ) f'( x ) +_du da" x fix ) = - do to ) do " d ( 1 + x " ) f(x ) - E Cr d ( 1+x " )- dant f'lx ) dx" t' (x ) + cr ( 1+29 84 - 1 dX"7 f' ( x ) + ( 2 x ( 1 + x * ) d " - dam-i f' ( x ) = Co (Itx2 ) finto ( x ) + (" ( 2 x ) fit ( x ) + ( 2 ( 2 ) + () ( x ) = (1+ x " ) fintz x ) + n(2x) font"(x ) + nen- dan xtla ) = 2 in-r Cr dx x d damer f(x ) Jan fix ) + (, dx x ( f(x ) (* ( x ) + 1 { ( (x ) ( 1 + x x ) flu +2 ( x ) + n( 2x ) fit ( x ) + nine ) 2 2 f" ( x ) + x fint ( x ) tuf" ( x ) = 0 ( Itx ' ) fin+ 2) in (x ) +