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m238sp06t1z - Dierential Equations test one Tuesday x 1...

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Differential Equations test one Tuesday, April 25, 2006 1 . Given the function y = 4+ x 2 sin s 2 ds ( a ) Calculate y (2). ( b ) Calculate dy dx . 2 . Is the function y = x + 1 x a solution of dy dx + y 2 = 3+ x 2 ? 3 . Determine whether each of the following equations is linear or non-linear. ( a ) dy dt = t - y 2 ( b ) t dy dt = e t y - cos t 2 ( c ) y dy dt = 4 t - y 4 . Which of the following equations is exact? ( a ) 3 x 2 - 4 y dx +(2 y - 4 x ) dy = 0 ( b ) dy dx + x 2 y = x 5 . Given the initial value problem t dy dt + 3 t - 2 y = 4 t + 1 and y (3) = 5 determine the largest t -interval over which a unique solution is guaranteed to exist. 6 . Given the initial value problem dy dx = y 3 + x 3 y and y (3) = 0 can one be certain that a unique solution exists?
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page two 7 . Use Euler’s method (i.e. the tangent-line method) with step-size h = 1 / 2 to fill in the following table for the initial value problem dy dx = 2 x - y 2 and y (1) = 2 x y dy dx approximate Δ y 1 2 1 . 5 8 . Solve each of the following differential equations or initial value problems:
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