>>11395994multiplying by differentials often is a chain rule in disguise. Consider the fiddling done in the pic related. Each line could be translated in a more sane language (at least for me) as:
Line 1: y'= f(x)g(y)
#remember, we can regard y as a function of x. this is a case of equation where you can separate the expression into the product of some f of x and some g of y
Line 2: y'/g(y)= f(x)
now take the integral with respect to x:
Line 3: integral ( y'/g(y)) dx = integral (f(x) dx) + c
#c as an arbitrary constant, since if you take a derivative of both sides it will vanish
then if you are not sure how to proceed as the text potrays this as if left side was integrated with respect to y, right side with respect to x, notice that if you make a substitution on the LHS of u= y and du = y' dx
here we use the fact that y is a function of x,
you get integral (1/g(u) du) = integral(f(x)dx) + c.
Most of the time differential fiddling CAN be translated in that way into more familiar language you learned in calculus class, it is just that the notation they are using works as the inner workings are the same. Do not give up, quick search on stackexchange will bring anwsers most of the time. I also dislike the notation.
