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HW AnswerGuide1 - Answer Guide 1 Math 427K Unique Number...

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Answer Guide 1 Math 427K: Unique Number 55160 Wednesday, August 31, 2011 1. Suppose that the area bounded by a smooth curve y ( x ), the x -axis, the y -axis, and the vertical line through x equals twice the curve’s arc length between 0 and x . Find the di ff erential equation satisfied by the curve. Solution: Writing x 0 y x ) dˆx = 2 x 0 1 + ( y x )) 2 dˆx and applying the fundamental theorem of calculus, one obtains y ( x ) = 2 1 + ( y ( x )) 2 . Simplifying, we get ( y ( x )) 2 = y ( x ) 2 4 1 . ( ) 2. A curve in the x, y -plane is defined by the condition that the sum of the x - and y -intercepts of its tangents always equals m . Find a di ff erential equation for the curve. Solution: Note that a correct answer here may be written in more than one form. The tangent line so a smooth curve y ( x ) at an arbitrary point ( a, b ) has the form y b = y ( a )( x a ) . The y -intercept, found by setting x = 0, is y int ( a, b ) = b ay ( a ) . The x -intercept, found by setting y = 0, is x int ( a, b ) = a b y ( a ) . (You may assume that y ( a ) is never zero, since otherwise the x -intercept does not exist.) Note that we have found y int and x int as functions of the point ( a, b ). Since ( a, b ) was arbitrary, we can substitute x = a , y = b , and dy dx = y ( x ) to get a nicer formula m = x 1 dy dx + y 1 dx dy . ( ) To get this formula, I also used the consequence of the chain rule that dy dx dx dy = 1. Another
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