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SeparableEquations

SeparableEquations - Separable Equations Definition An...

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Separable Equations Definition An O.D.E. of the form F(x) G(y) dx + f(x) g(y) dy = 0 is called an equation with variable separable or a separable equation . Example : The O.D.E. (x 3 ! 3) F(x) ! " #### y 2 G(y) !" dx ! x f(x) !" (y + 1) g(y) ! " ### dy = 0 is separable. In general a separable equation is non-exact but it has an obvious integrating factor, p(x,y) = 1 f(x)G(y) when we multiply the equation by p(x,y), we get the essential equivalent exact equation F(x) f(x) dx + g(y) G(y) dy = 0 since ! ! y F(x) f(x) " # $ % & = 0 = ! ! x g(y) G(y) " # $ % & . Since the first term of the equation depends on x only, we can integrate it with respect to x. Since the second term depends on y only, we can integrate it with respect to y. F(x) f(x) dx + g(y) G(y) dy = c ! ! and we obtain the one-parameter family of solutions of the separable equation. Remark : Notice that we must separate the variables before integration. Remark : To separate the variables, we divide by f(x)G(y). Since division by zero is not allowed, we must assume that neither f(x) = 0 or G(y) = 0.
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