Assignment2-solution-public

Assignment2-solution-public - Math 201 Fall 2011 Assignment...

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Unformatted text preview: Math 201 Fall 2011 Assignment #2 Solutions October 1, 2011 Grade problems 1 and 2, 10 marks each, 20 total for this assignment. 1. y = 1 /t + u , so, y =- 1 t 2 + u =- 1 t 2- 1 t ( 1 t + u ) + ( 1 t + u ) 2 =- 1 t 2 + u t + u 2 Thus, the equation for u is u = u/t + u 2 , which is a Bernoulli equation. Let w = 1 /u , w =- u u 2 =- u/t + u 2 u 2 =- 1 t w- 1 this is a linear equation, i.e., w + w/t =- 1 . The integrating factor is ( t ) = e (1 /t ) dt = e ln t = t Multiply ( t ) on both sides, ( w ) = w + w t =- =- t Integrate on both sides, w = tw =- t 2 / 2 + C , isolate w , w =- t 2 + C t . Now substitute in u , u = 1 w = 1- t/ 2 + C/t = 2 t- t 2 + C 1 where C 1 = 2 C . So, y = 1 t + u = 1 t + 2 t- t 2 + C 1 1 2. The right hand side f ( x,y ) = y x (1 + ln y- ln x ) = y x 1 1 + ln( y/x ) To check the homogeneity, f ( kx,ky ) = ky kx 1 1 + ln ky kx = y x 1 1 + ln( y/x ) = f ( x,y ) so the rand hide side is homogeneous to degree 0. To solve it, let u = y/x , i.e.,i....
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Assignment2-solution-public - Math 201 Fall 2011 Assignment...

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