ch02_2 - Ch 2.2: Separable Equations In this section we...

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Ch 2.2: Separable Equations In this section we examine a subclass of linear and nonlinear first order equations. Consider the first order equation We can rewrite this in the form For example, let M ( x , y ) = - f ( x , y ) and N ( x , y ) = 1. There may be other ways as well. In differential form, If M is a function of x only and N is a function of y only, then In this case, the equation is called separable . 0 ) , ( ) , ( = + dx dy y x N y x M 0 ) , ( ) , ( = + dy y x N dx y x M 0 ) ( ) ( = + dy y N dx x M ) , ( y x f dx dy =
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Example 1: Solving a Separable Equation Solve the following first order nonlinear equation: Separating variables, and using calculus, we obtain The equation above defines the solution y implicitly. A graph showing the direction field and implicit plots of several integral curves for the differential equation is given above. 1 1 2 2 - + = y x dx dy ( 29 ( 29 ( 29 ( 29 C x x y y C x x y y dx x dy y dx x dy y + + = - + + = - + = - + = - 3 3 3 1 3 1 1 1 1 1 3 3 3 3 2 2 2 2
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This note was uploaded on 04/11/2011 for the course MECHANICAL MAT 219 taught by Professor Benjaminwalter during the Fall '10 term at Middle East Technical University.

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ch02_2 - Ch 2.2: Separable Equations In this section we...

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