This is used in instances where it is a first order DE with two functions The strategy is to separate the two functions before integration.
\(\dfrac{dy}{dx} = 5x\,y\)What is the general solution?
\(dx = 5x\,y\,dy\)
\(\dfrac{1}{5x}\,dx = y\,dy\)
\(\int \dfrac{1}{5x}\,dx = \int y\,dy\)
\(\ln |5x| = \dfrac{y^{2}}{2} + C\)
\(e^{\ln |5x|} = e^{\dfrac{y^{2}}{2}} + e^{C}\)
\(5x = e^{\dfrac{y^{2}}{2}} + e^{C} \)
\(x = \dfrac{1}{5}e^{\dfrac{y^{2}}{2}} + e^{C}\)
