# PP 10.3_1 - Separable Equations Definition A separable...

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Separable Equations

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Definition A separable equation is a first-order differential equation that can be written in the form ) ( ) ( y f x g dx dy = So we want to be able to write the right hand side of the equation as the product of a function of x and a function of y .
How to Solve xy dx dy 2 = The method for solving a separable equation is called separation of variables because that is exactly what is done before integrating to find the solution set. Example: dx x y dy 2 = Now we can integrate both sides with respect to the appropriate variable.

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= = dx x y dy dx x y dy 2 2 C x y + = 2 | | ln Now we need to solve for y to get an explicit solution to the differential equation. C x e y + = 2 | | C x x C x C e A Ae e e y e e y ± = = ± = = , | | 2 2 2 Using properties of exponents and of absolute value: We see that the solution is an infinite family of functions.
Example ) ln( 2 equation al differenti the Solve y xy y = = ) ln(

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## PP 10.3_1 - Separable Equations Definition A separable...

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