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Unformatted text preview: GUIDELINES FOR SEPARABLE FIRST ORDER EQUATIONS Given the separable differential equation y = f ( x ) g ( y ) , the steps to find a solution are the following: Step 1: Find all constant (equilibrium) solutions by solving the equation g ( y ) = 0. Step 2: Separate variables and integrate to find an implicit solution with possible restric tions on the additive constant. Step 3: If possible, find explicit solutions from the implicit solution. Determine their intervals of definition. Step 4: If initial data are given, use them in Step 2 to determine the additive constant and again in Step 3 to insure that you have the desired explicit solution. Example: Consider the differential equation dy dx = x x y 2 y + x 2 y . 1. x x y 2 y + x 2 y = 0 if and only if x (1 y 2 ) = x (1 y ) (1 + y ) = 0 so the equilibrium solutions are y ( x ) ≡ 1 , and y ( x ) ≡  1 . 2. Separating variables in the equation we get y (1 y ) (1 + y ) dy dx = x 1 + x 2 or 1 2 1 1 y 1 1 + y dy dx = x 1 + x 2 ....
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 Spring '08
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 Calculus, Equations, Elementary algebra, Constant of integration

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