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Unformatted text preview: MATHEMATICS 3161: Fall 2009 (2009.9  2009.12) Assignment # 2 (Due Date: Oct. 5) 1. Find all singular points of the given equation and determine whether each one is regular or irregular. a) (1 x 2 ) 2 y + x (1 x ) y + (1 + x ) y = 0 b) ( x 2 x 2) y + ( x + 1) y + 2 y = 0, c) x 2 y + 2( e x 1) y + ( e x cos x ) y = 0 Solution : a) p ( x ) = x (1 x )(1 + x ) 2 , q ( x ) = 1 (1 + x )(1 x ) 2 both are rational functions. x = 1 are singular points. lim x 1 ( x 1) p ( x ) = 1 4 , lim x 1 ( x 1) 2 q ( x ) = 1 2 , lim x 1 ( x + 1) p ( x ) = , therefore, x = 1 is a regular singular point, x = 1 is an irregular singular point. b) p ( x ) = 1 x 2 , q ( x ) = 2 ( x + 1)( x 2) both are rational functions. x = 1 , 2 are singular points. lim x 1 ( x + 1) p ( x ) = 0 , lim x 1 ( x + 1) 2 q ( x ) = 0 , lim x 2 ( x 2) p ( x ) = 1 , lim x 2 ( x 2) 2 q ( x ) = 0 , therefore, x = 1 , 2 are regular singular points. c) p ( x ) = 2( e x 1) x 2 , q ( x ) = e x cos x x 2 , they are not rational functions. x = 0 is singular point. xp ( x ) = 2( e x 1) x = 2 summationdisplay n =1 x n 1 n ! = 2 summationdisplay n =0 x n ( n + 1)! is convergent for all x , ( R = ), x 2 q ( x ) = e x cos x is convergent for all x since both e x and cos x are convergent for all x , so x = 0 is a regular singular point. 2. Determine the general solution of the given differential equation that is valid in any interval not including the singular point. a) x 2 y 5 xy + 9 y = 0 b) ( x 2) 2 y + 5( x 2) y + 8 y = 0 c) x 2 y 4 xy + 4 y = 0 Solution : There are all Euler equations. a) the characteristic equation is r ( r 1) 5 r + 9 = r 2 6 r + 9 = ( r 3) 2 = 0 then r = 3 (equal). So the general solution is y = c 1 x 3 + c 2 x 3 ln  x  , ( x negationslash = 0). b) the characteristic equation is r ( r 1) + 5 r + 8 = r 2 + 4 r + 8 = 0 r = 4 16 32 2 = 2 2 i (complex) , x = 2 Then the general solution is y =  x 2  2 ( c 1 cos 2 ln  x 2  + c 2 sin ln2  x 2  ) , ( x negationslash = 2)....
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 Spring '10
 schoutz
 Math

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