The differential equation which represents the family of plane curves y = exp(cx) is:

y' = cy xy' - log y = 0 x log y = yy' y log y = xy'

y log y = xy'

Take log on both sides  ln y = cx     … (i) Differentiate (i) w. r. t. x  ; $\frac{1}{y}\frac{dy}{dx}=c$   … (ii)      Using (i) and (ii), we have  y log y = xy'