First order bdifferential equations/b of the form (\dfrac{dy}{dx}~=~f(x)g(y) ) can be solved by bseparating the variables/b.

(\dfrac{dy}{dx} = f(x)g(y) )

(dy) and (dx) can be treated as variables and manipulated the usual way.

(\dfrac{1}{g(y)} dy = f(x)dx )

Integrate both sites with respect to (y) and (x) respectively.

(\displaystyle\int{\dfrac{1}{g(y)}} dy = \displaystyle\int{f(x)} dx )

The bgeneral solution/b to a differential equation will be in the form:

(y = F(x) + c )

If you know one point on the curve, (c) can be solved and a bparticular solution/b can be obtained.

[b]uSolving differential equations[/u]/b

(\dfrac{dy}{dx} = f(x)g(y) \implies \displaystyle\int{\dfrac{1}{g(y)}} dy~=~\displaystyle\int{f(x)} dx )