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A first order differential equation is called seperable if it can be written as

$ {{dy \over dx} = f(x)g(y)} $.

In this case, the equation can informally be written as

$ {g(y) dy = f(x) dx} $.

By taking the integral of both sides, the solution can be written as

$ {F(y(x)) = G(x) + C} $

where,

$ {F(y) = \int f(y) dy, G(x) = \int g(x) dx} $

and  is a constant of integration.

Example

$ {{dy \over dx} = y sin(x)} $

$ {\Rightarrow {1 \over y} dy = sin(x) dx} $

$ {\Rightarrow \int {1\over y} dy = \int sin(x) dx} $

$ {\Rightarrow ln|y| = -cos(x) + C} $

$ {\Rightarrow y = e^{-cos(x) + C}} $

$ {\Rightarrow y = De^{-cos(x)}} $

External References

http://en.wikipedia.org/wiki/Seperation_of_variables