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Unformatted text preview: Â¿ and N ( tx ,ty )=( tx ) 2 âˆ’( tx ) ( ty )= t 2 x 2 âˆ’ t 2 ( xy )= t 2 ( x 2 âˆ’ xy ) Â¿ t 2 N ( x , y ) both of coefficients are homogeneous functions of degree 2. If we let y = ux ( u = y x ) then dy = duâˆ™ x + udx ( x 2 + u 2 x 2 ) dx + ( x 2 âˆ’ x 2 u ) [ duâˆ™ x + u dx ]= ( 1 + u 2 ) dx +( 1 âˆ’ u ) [ duâˆ™ x + u dx ]= ( 1 + u 2 ) dx +( 1 âˆ’ u ) udx +( 1 âˆ’ u ) xdu = ( 1 + u 2 + u âˆ’ u 2 ) dx +( 1 âˆ’ u ) udx = ( 1 + u ) dx +( 1 âˆ’ u ) udx = dx x + ( 1 âˆ’ u ) du ( 1 + u ) = dx x + ( âˆ’ 1 + 2 ( 1 + u ) ) du = After integrating the last line gives ln  x âˆ’ u + 2ln  1 + u = ln âˆ¨ C âˆ¨ Â¿ ln  x âˆ’ y x + 2ln  1 + y x  = ln âˆ¨ C âˆ¨ Â¿ y x = ln  x C  + ln  ( x + y ) x 2 2  y x = ln  ( x + y ) x 2 2 x C  , y x = ln  ( x + y ) Cx 2  or y = x ln  ( x + y ) Cx 2  ....
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 Fall '15
 Linear Algebra, Algebra, Equations, Trigraph, dx, homogeneous functions

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