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Unformatted text preview: MAP2302, FALL 07 EXAM1 To receive full credit you must carefully explain your answers 1. Determine those m so that φ ( x ) = x m is a solution of x 2 y 00- xy- 3 y = 0 . Problem 21 b section 1.2. With the choice of φ , note that φ = mx m- 1 and φ 00 = m ( m- 1) x m- 2 . Substituting into the differential equation gives, m ( m- 1) x m- mx m- 3 x m = 0 . Thus, x m is a solution if and only if m 2- 2 m- 3 = 0. Thus, m = 3 ,- 1. 2. Does the relation e xy + y = x- 1 determine an implicit solution of y = e- xy- y e- xy + x ? Problem 11 section 1.2. Differentiating the expression implicitly with respect to x gives, e xy ( xy + y ) + y = 1 . Solving for y we find, (1 + xe xy ) y = 1- ye xy which, after rearranging, gives, y = e- xy- y e xy + x . Thus the expression does define an implicit solution to the given dif- ferential equation. 3. Draw some isoclines for y = 2 x 2- y . What is the slope of the solution passing through (0 , 1) at the point (0 , 1)?...
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This note was uploaded on 12/15/2011 for the course MAP 2302 taught by Professor Tuncer during the Spring '08 term at University of Florida.
- Spring '08