I'm quite a novice with differential equations and such and english isn't my first language so I ask to please explain everything in simple english terms.
I am currently studying differentiation equations in school and one of the tricks is "separation of variables". The trick is very simple and I get how its used. You replace $y'$ with $dy/dx$ and juggle the equation around until all the $y$'s are on one side and all the $x$'s are on the other. Then you can integrate
However what I do NOT understand is how this is necessary in a mathematical sense.
$y'$ is not $y$. It is the function of change for $y$. So why can't I just integrate right away? I know it won't work but I don't get why it wouldn't.
Let's say we got the following function:
$$y' = -\frac{x}{y}$$
I understand that if I don't separate the variables that $y$ will remain in the solution so essentially the solution of $y$ will have $y$ in it, but in a mathematical sense I still don't get the connection. Is it really just a cheap trick or is there actual logic behind whats happening here other than just "doing it to get rid of $y$".
Best Answer
I agree that this method can seem a bit mysterious at first so let's try to understand what is really going on here.
First let's start by considering a relation between $y(x)$ and $x$ on the form $F(y) = G(x)$. Taking the derivative gives us a separable ODE $f(y)y' = g(x)$ where $f = F'$ is the derivative of $F$ and $g = G'$ is the derivative of $G$.
Separation of variables is nothing but this in reverse. Given a ODE that is separable, i.e. it's on the form $f(y)y' = g(x)$, then applying the method we are lead to solve
$$\int f(y)dy = \int g(x)dx$$
which we know from above leads to the relation $F(y) = G(x)$. All you are doing when applying separation of variables is to try to find $F$ and $G$ from $f$ and $g$. The seemingly dippy process of writing $y' = \frac{dy}{dx}$ and treating this as a fraction to get the integral equation above is just a simple way of remembering how to do this.