I have many questions I'd like to ask today. I'm currently studying for my A-levels which is a qualification mainly based in England. I'm taking A-level Further Maths, which involves the study of First and Second Order Differential Equations. It seems we are just being taught to apply recipes to these types of equations and are given no intuition as to why these things even work? This isn't just for teaching in England, I've watched many American lectures and lecture notes and the same thing is done.
Let's take a look at Non-Linear Separable First Order Differential Equations.
We have defined these equations to take the form $$ N(y)\frac{dy}{dx} = M(x)$$
We are taught to solve these equations by multiplying $\ dx $ to both sides and then integrate by their respective variables. Like so:
$$ \int N(y)\ dy = \int M(x) \ dx $$
I have just accepted this for some time, but I'd like to get to the bottom of wtf is going on. We are taught from Year 12/Calculus 1 days that $\frac{dy}{dx} $ is not a ratio but can be treated like one in many cases. Ok... then could you explain really what we just did when solving the above differential equation? I can accept the fact that we can treat $\frac{dy}{dx}$ as a ratio but it's not really a ratio. So, can we be told what we are really doing? Is it some kind of hidden secret? I just want to know why we multiplied both sides by $\ dx$, what are we actually doing?
I can assume that the reason we are not told is because it involves higher level math, that someone taking Calculus would not understand. But will actual reasoning behind why these recipes work be exposed to us during University math courses?
