Separable Differential Equations

  • Thread starter Bashyboy
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  • #1
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Main Question or Discussion Point

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

[itex]\frac{dy}{g(y)} = h(x)dx[/itex]

integrating

[itex]G(y) = H(x) + c[/itex]

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

Could someone please help?
 
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Answers and Replies

  • #2
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The justification is that dividing by g(y) results in

[tex]\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}[/tex]

which then, supposing (chain rule) that for some function G(y)
[tex]\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},[/tex]

can be integrated to

[tex]\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,[/tex]

the result you get if you "divide by dy". The steps you made are commonly used as they are often considered more intuitive.
 
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  • #3
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You can think of the steps that seem incorrect as mnemonics to obtain a correct result. The justification is the one given above.
 

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