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Unformatted text preview: September 7, 2011 3-1 3. Separable differential Equations A differential equation of the form dy dx = f ( x,y ) is called separable if the function f ( x,y ) decomposes as a product f ( x,y ) = φ 1 ( x ) φ 2 ( y ) of two functions φ 1 and φ 2 . Proceding formally we can rewrite this as dy dx = φ 1 ( x ) φ 2 ( y ) dy φ 2 ( y ) = φ 1 ( x ) dx. Using the second formula, we can integrate both sides to get Z y dy φ 2 ( y ) = Z x φ 1 ( x ) dx + C as the general solution. Note that this is an implicit relation between y ( x ) and x . Indeed, the last integral formula has the form F ( y ( x )) = G ( x ) + C for some functions F and G . To find y ( x ) as a function of x we would have to solve this implicit relationship. This is frequently hard to do, so we will leave the solution in implicit form. A more general version of this is the d.e. M ( x ) dx + N ( y ) dy = 0 (1) We say that the general solution to this d.e. is an expression September 7, 2011 3-2 f ( x,y ) = C where f x = M (...
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