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}


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

and  is a constant of integration.


{{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

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