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# Q2ts - Quiz II y 1 a The differential form =(ln y x 1)dx(ln...

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Quiz II 1. a) The differential form ω = (ln y + y x - 1) dx + (ln x + x y - 1) dy is exact in any rectangle contained in the first quadrant (where x > 0 and y > 0). Find a function F ( x, y ) defined in the first quadrant such that dF = ω . Solution . Following the standard procedure F ( x, y ) = Z (ln y + y/x - 1) dx + h ( y ) = x ln y + y ln x - x + h ( y ) ∂F ∂y = x/y + ln x + h 0 ( y ) So we need h 0 ( y ) = - 1, which gives h ( y ) = - y + C and F ( x, y ) = x ln y + y ln x - x - y + C. The level curves of the function F ( x, y ) that you found in part a) are solutions to a first order differential equation. Write down that differential equation. Solution . We have ω = 0 when y = y ( x ) is a solution to the differential equation. So 0 = (ln y + y/x - 1) dx +(ln x + x/y - 1) dy dx dx = [(ln y + y/x - 1)+(ln x + x/y - 1) dy dx ] dx So the differential equation is 0 = (ln y + y/x - 1) + (ln x + x/y - 1) dy dx or in normal form dy dx = - ln y + y/x - 1 ln x + x/y - 1 . Of course, you could get that just by setting ω = 0, dividing by dx and then solving for dy dx . That “dividing by dx ” step does not make sense, but the notation is designed so that it leads to the right answer!

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Q2ts - Quiz II y 1 a The differential form =(ln y x 1)dx(ln...

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