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Unformatted text preview: Separable Equations We have found a strategy for solving first-order linear equations, and would now like to consider a particular class of first-order nonlinear equations. We will again make use of exact derivatives. Definition A differential equation that can be put in the form dy dx = g ( x ) f ( y ) is called separable . To solve separable equations, we can multiply by f ( y ) to get (1) f ( y ) dy dx = g ( x ) and observe that the LHS looks like the result of applying the chain rule to some function F ( y ) where F is an antiderivative for f . In particular, we have from the chain rule d dx F ( y ) = F ( y ) dy dx = f ( y ) dy dx . So we can rewrite the LHS of (1) as an exact derivative to get d dx F ( y ) = g ( x ) and integrating both sides gives F ( y ) = Z g ( x ) dx. If G is an antiderivative for g then we have (2) F ( y ) = G ( x ) + C. Now from equation (2), we may or may not be able to solve for y as an explicit function of x . But even if we cant, equation (2) still defines...
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