Separable Differential Equations

[Bashyboy]

I have read that, if you given a differential equation $\frac{dy}{dx} = f(x,y)$, and can write it in the form $\frac{dy}{dx} = h(x)g(y)$, then you can proceed with the following steps:

$\frac{dy}{g(y)} = h(x)dx$

integrating

$G(y) = H(x) + c$

Why are these steps vaild? I thought that one was not supposed to regard $\frac{dy}{dx}$. I have heard that you can regard it as a fraction, because, before taking the limit, you can manipulate the fraction $\frac{\Delta y}{\Delta x}$.

Could someone please help?

 

[DeIdeal]

The justification is that dividing by g(y) results in

$$\frac{\mathrm{d}y}{\mathrm{d}x}=h(x)g(y)\implies h(x)=\frac{1}{g(y)}\frac{ \mathrm{d} y}{\mathrm{d}x}$$

which then, supposing (chain rule) that for some function G(y)
$$\frac{\mathrm{d}}{\mathrm{d}y}G(y)=\frac{1}{g(y)}\implies\frac{\mathrm{d}}{\mathrm{d}x}G(y)=\frac{1}{g(y)}\frac{\mathrm{d}y}{\mathrm{d}x},$$

can be integrated to

$$\int h(x) \mathrm{d} x = \int \frac{1}{g(y)}\frac{\mathrm{d}y}{\mathrm{d}x}\mathrm{d}x = \int (\frac{\mathrm{d}}{\mathrm{d}x}G(y)) \mathrm{d}x= G(y) = \int (\frac{\mathrm{d}}{\mathrm{d}y}G(y))\mathrm{d}y=\int \frac{1}{g(y)} \mathrm{d}y,$$

the result you get if you "divide by dy". The steps you made are commonly used as they are often considered more intuitive.

 

[hellofolks]

You can think of the steps that seem incorrect as mnemonics to obtain a correct result. The justification is the one given above.