practice(exam1)MA262fall2011

practice(exam1)MA262fall2011 - PRACTICE PROBLEMS FOR EXAM 1...

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PRACTICE PROBLEMS FOR EXAM 1 - MA 262 FALL 2011 INSTRUCTOR: RAPHAEL HORA 1. Solve (1 - y 5 ) dy dx = xe x 2 , y (0) = 0 . 2. Solve dy dx = 2 xy x - 1 , y (0) = 1. 3. Solve dy dx = 2 y (1 - y ) x . 4. Verify that y = - x solves dy dx = x + 3 y x - y and compute other solutions to this problem. See the Existence and Uniqueness Theorem to see the reasons why this problem may not have a unique solution. Where are these solutions valid? 5. Solve dy dx = 3 y 2 - x 2 2 xy . Make v = y/x, so we have v + x dv dx = 3 v 2 - 1 2 v 2 v v 2 - 1 dv = 1 x dx To compute the integral in the left-hand side, make u = v 2 - 1 , so du = 2 vdv. Hence ln | v 2 - 1 | = ln | x | + C ln ± ± ± ± y 2 x 2 - 1 ± ± ± ± = ln | x | + C ln | y 2 - x 2 | - ln x 2 = ln | x | + C ⇒ | y 2 - x 2 | = kx 3 . Recall that ln x 2 = 2 ln | x | and ln ± ± ± ± y 2 x 2 - 1 ± ± ± ± = ln ± ± ± ± y 2 - x 2 x 2 ± ± ± ± = ln | y 2 - x 2 | - ln x 2 . 6. A tank contains 4L (liters) of milk in which is dissolved 400g (grams) of Nesquik.
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practice(exam1)MA262fall2011 - PRACTICE PROBLEMS FOR EXAM 1...

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