I'm currently reading Differential Equations for Dummies, and this is what it says on pg 60.
I wasn't too sure whether "flipping fractions" still satisfy the equation, and I searched online (including this site), but it seems that most people say flipping fraction is not correct. At the same time, there does seem to be a specific method that allows you to "flip fractions" correctly. Could someone help me understand how? (i.e. how to get from the first eq shown to the second eq.)
The thing that justifies this is the chain rule.
An example occurring in first-year calculus can illustrate this. Suppose you know that $$ \frac{dy}{dx} = \frac d{dx} \tan x = \sec^2 x = 1+\tan^2 x = 1+y^2. $$ $$ \frac{dy}{dx} = 1+y^2. $$ $$ \frac{dx}{dy} = \frac 1 {1+y^2}. $$ $$ \frac d{dy} \arctan y = \frac 1 {1+y^2}. $$ So how is this the chain rule? $$ x = \arctan y = \arctan(\tan x) \tag 1 $$ Differentiating both sides of $(1)$ yields $$ 1 = {} \overbrace{ \frac d{dx} \arctan y = \big(\arctan'y\big)\cdot \frac{dy}{dx} }^\text{chain rule} $$ Therefore $$ \frac 1 {dy/dx} = \arctan'y. $$
(And then $dy/dx = \sec^2 x = 1+\tan^2 x = 1+y^2.$)