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Quiz (24) - PRINT Your Name Quiz for June 5 2012 The quiz...

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PRINT Your Name: Quiz for June 5, 2012 The quiz is worth 5 points. Remove EVERYTHING from your desk except this quiz and a pen or pencil. SHOW your work. Express your work in a neat and coherent manner. BOX your answer. 1. Solve yy + x = r x 2 + y 2 . Express your answer in the form y ( x ) . Check your answer. This is a homogeneous problem. Divide both sides by x to write the problem as y x y + 1 = R 1 + p y x P 2 . Let v = y x . In other words, xv = y . Take the derivative with respect to x to see that xv + v = y . We must solve v ( xv + v ) + 1 = r 1 + v 2 . We must solve xv dv dx = r 1 + v 2 - v 2 - 1 . We must solve v dv 1 + v 2 - v 2 - 1 = dx x . Integrate both sides. Let w = 1 + v 2 . It follows that dw = 2 vdv . We must solve 1 2 i dw w - w = ln | x | + C. We have ln | x | + C = 1 2 i dw w (1 - w ) . Let u = w . We have du = 1 2 w 1 / 2 dw . ln | x | + C = i du 1 - u = - ln | 1 - u | = - ln | 1 - w | = - ln | 1 - r 1 + v 2 | = - ln v v v v v 1 - R 1 + p y x P 2 v v v v v = - ln v v v v v x - r x 2 + y 2 x v v v v v = - ln v v v x
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