S22 - 2.2 Separable Equations A first order differential...

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2.2 Separable Equations A frst order differential equation y ± = f ( x, y )i sa separable equation iF the Function f can be expressed as the product oF a Function oF x and a Function oF y . That is, the equation is separable iF the Function f has the Form f ( x, y )= p ( x ) h ( y ) . where p and h are continuous Functions. The solution method For separable equations is based on writing the equation as 1 h ( y ) y ± = p ( x ) or q ( y ) y ± = p ( x ) (1) where q ( y )=1 /h ( y ). OF course, in dividing the equation by h ( y ) we have to assume that h ( y ) ± = 0. Any numbers r such that h ( r )=0 may result in singular solutions oF the Form y = r . IF we write y ± as dy/dx and interpret this symbol as “differential y ” divided by “differential x ,” then a separable equation can be written in differential Form as q ( y ) dy = p ( x ) dx. This is the motivation For the term “separable,” the variables are separated. Solution Method for Separable Equations Step 1. IdentiFy: Can you write the equation in the Form (1). IF yes, do so. In expanded Form, equation (1) is q ( y ( x )) y ± ( x p ( x ) . Step 2. Integrate this equation with respect to x : ± q ( y ( x )) y ± ( x ) dx = ± p ( x ) dx + CC an arbitrary constant which can be written ± q ( y ) dy = ± p ( x ) dx + C by setting y = y ( x ) and dy = y ± ( x ) dx . Now, iF P is an antiderivative For p , and iF Q is an antiderivative For q , then this equation is equivalent to Q ( y P ( x )+ C. (2) INTEGRAL CURVES Equation (2) is a one-parameter Family oF curves called the integral curves oF equation (1). In general, the integral curves defne y implicitly as a Function oF x . These curves are solutions oF (1) since, by implicit differentiation, d dx [ Q ( y )] = d dx [ P ( x )] + d dx [ C ] q ( y ) y ± = p ( x ) .
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This note was uploaded on 06/20/2008 for the course M 427K taught by Professor Fonken during the Spring '08 term at University of Texas at Austin.

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S22 - 2.2 Separable Equations A first order differential...

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