Winter 2006 - Reynold's Class - Exam 1 (Version B)

# Winter 2006 - Reynold's Class - Exam 1 (Version B) - Exam 1...

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1. [ 24 points ] Determine whether each of the following equations is linear, separable, exact, or none of the above – in other words, identify a solution approach for the problem. Some of these may satisfy multiple labels – pick one. Do not solve the diﬀerential equation. a. [ 8 pts ] dy dx = 1 4 x (1 + x 2 ) y - 3 Solution: We re-write the equation as 4 y 3 dy dx - x (1 + x 2 ) = 0 . This is in separable form, so the equation is separable . Moreover, we note that M y ( x ) = N x ( y ) = 0 , so it is also exact . It is not linear. b. [ 8 pts ] dy dx = 6 x - 2 x - 2 y 4 - 2 x - 1 Solution: We multiply through by the denominator of the right-hand side to get (4 - 2 x - 1 ) dy dx + (6 x - 2 x - 2 y ) = 0 This is in exact form, but ﬁrst we must check that it is exact. Diﬀerentiating the ﬁrst term by x and the second by
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Winter 2006 - Reynold's Class - Exam 1 (Version B) - Exam 1...

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