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Unformatted text preview: Quiz II 1. a) The differential form = (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 = . 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 ( y ) So we need h ( 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 = 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 ....
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