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.