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**Unformatted text preview: **2.2: SEPARABLE EQUATIONS KIAM HEONG KWA A separated equation is a first-order equation of the form (1) M ( x ) + N ( y ) dy dx = 0 , where M ( x ) and N ( y ) are continuous functions on some (open) inter- vals. It is understood that x is the independent variable and y is the unknown function. Such an equation can be solved immediately by integrating both sides of the equation with respect to the independent variable x . That is, (2) Z M ( x ) + N ( y ) dy dx dx = c, where c is an integration constant. Using the rule of substitution, the last equation can also be written as (3) Z M ( x ) dx + Z N ( y ) dy = c. This equation defines the integral curves, i.e., the graphs of solutions, of (1), as well as diminishes the distinction between independent variable x and the unknown y . Hence the separable equation is sometimes presented in the differential form (4) M ( x ) dx + N ( y ) dy = 0 . A separable equation is an equation that can be transformed into a separated equation (with appropriate stipulations whenever necessary)....

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