Determine the values of the constants r and s such that i(x, y) = x rys is an integrating factor for the given differential equation. y(7xy2 + 6) dx + x(xy2 − 1) dy = 0

For the ODE to be exact, we require that , which we'll verify is not the case here.




So we distribute an integrating factor across both sides of the ODE to get



Now for the ODE to be exact, we require , which in turn means



Suppose . Then substituting everything into the PDE above, we have






so that our integrating factor is . Our ODE is now



Renaming and to our current coefficients, we end up with partial derivatives




as desired, so our new ODE is indeed exact.

Next, we're looking for a solution of the form . By the chain rule, we have



Differentiating with respect to yields




Thus the solution to the ODE is