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**Unformatted text preview: **M y = M ( x ) y = 0 and N x = 0. Since they are the same, any separable equation is also exact (since it can be written in the above form). 2.5.17 (a) Solve the Gompertz equation dy dt = ry ln K y , subject to the initial condition y (0) = y . Separating gives us Z dy y ln K y = Z r dt. For the integral on the left, if we let u = ln( K/y ) as suggested, then du =-dy/y , so the integral becomes-R u-1 du =-ln | ln( K/y ) | . So ln | ln( K/y ) | =-rt + C or ln( K/y ) = De-rt . 1 Finally, solving for the constant D , and then solving for y we get y ( t ) = Ke-ln( K/y ) e-rt = Ke ln( y /K ) e-rt . (b) If r = 0 . 71 per year, K = 80 . 5 10 6 kg, and y /K = 0 . 25, then y (2) = 57 . 580 10 6 . (c) y ( ) = 0 . 75 K when e ln(0 . 25) e-. 71 = 0 . 75 , i.e. when =-1 . 71 ln ln 0 . 75 ln 0 . 25 2 . 2148 . 2...

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