Differential Equations in Differential Form

Question

I'm reading on my analysis book that "the first order differential equation $y'=f(x,y)$ can be rewritten as $dy-f(x,y)dx = 0$ and that it is equivalent to the more general equation $L(x,y)dx+M(x,y)dy = 0$". Really I don't understand how can be made this "transformation" without thinking the derivative as a ratio of infinitesimal. Thanks in advance

Answer 1

Even though it might seem like so at first, you don't need to think of differentials as infinitesimals and derivatives as their ratios to make sense of those expressions.

You can define a differential $\text{d}y$ as the linear part of a variation $\Delta y = f(x+\Delta x)-f(x)$, or as an abstract mathematical object known as a differential form. Either way, you would get that (for one-variable differentiable functions)

$$\text{d}y = f'(x) \text{d}x = \frac{\text{d}y}{\text{d}x}\text{d}x$$

For a more in-depth explanation, you can check out my answer to this question here: When can we not treat differentials as fractions? And when is it perfectly OK?; or any other answer there for that matter.