According to Ince [pg. 531] the first known use of integrating factors to solve a differential equation was by Fatio de Duiller in June of 1687. He was solving the equation $$3xdy - 2ydx = 0$$ which we would write in standard form (using the prime notation) as

For this equation the integrating factor is:
After multiplying both sides by the integrating factor and unapplying the product rule we get the new differential equation:

Integrating both sides we get the algebraic equation $=$
Solving for y, the solution to the differential equation is y = (using C as the constant)

[Ince] Ince E L, Ordinary Differential Equations, Longmans, Green and Co, London, 1927.