Solve the differential equation: (dy/dx) = 6xy^2

Start by recognising that this is a separable differential equation; it can be written with all of the x's on one side of the equals sign, and all of the y's on the other. The first step is to rearrange so that:

(1/y^2)(dy/dx) = 6x

Now integrating both sides with respect to x gives:

integral (1/y^2 ) dy = integral (6x) dx

Carry out the integral:

-1/y = 3x^2 + k

And rearrange to give:

y = -1/(3x^2 + k)