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In solving ordinary differential equations one technique is separation of variables. Lets say we have ${dy}/{dx} = xy$. Then we divide by $y$ so that the left side is in terms of $y$ and the right side of is in terms of $x$; $1/y * dy/dx = x$. If I'm correct so far then we simply integrate both side in terms of $x$ giving us $\int (1/y * dy/dx) dx = \int x dx $. Many people seems to think that we can simply eliminate the $dx$'s on the left to get $\int (1/y)dy$. This feels wrong to me because the $dy/dx$ is a fraction and can be treated as variables, but on the left the $dy$ at the end of the integral $\int dx$ is not a variable, its part of the notation. So can anybody explain this to me and point me to where I can read up more about the algebraic treatment of the $dx,dy$ etc variables.

Johan
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  • Multiply the equation by $dx$ and divide the equation by $y$. Then integrate. This is a classic case of separation of variables and is perfectly valid. – Bill Watts May 08 '22 at 07:22

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Notation can make the whole thing easier to understand. Note that $dy/dx$ is not a fraction but just notation itself.

With different notation, your problem becomes $ y’ = x y$. Then you can divide both sides by $y$. You end up with $ y’ / y = x $. Note integrate both sides with respect to the argument $x$. You end up with $\log( y ) = \frac{ x^2 }{ 2 } + C$ where $C$ is any constant.

NicNic8
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