let $g'(x)=\frac{1}{x^2};x \neq 0$-------(1)
since, $\int g'(x)dx= \left\{\begin{array}{ll}\frac{1}{x}+c_1;x>0\\ \mathbf{and} \\ \frac{1}{x}+c_2;x<0\end{array}\right.$-----(1.1)
because, g'(x) is a piece-wise function and the pieces of the integral of g'(x) can differ by a constant and still have the same derivative.
then why do we have,
$\int dy=\int ydx$
$\int \frac{dy}{y}=\int dx$-------(2)
$=>\left\{\begin{array}{ll}ln|y|=x+c_1;y>0\\ \mathbf{or} \\ln|y|=x+c_2;y<0\end{array}\right.$
$=>\left\{\begin{array}{ll}y=c_1e^x;y>0\\ \mathbf{or} \\y =c_2e^x;y<0\end{array}\right.$-------(3)
i know, that having or in (3) makes sense (because if we had and instead then, the resulting curve would not be a function and thus derivatives would not be defined at all.)
but if we follow the same method we used for integrating (1), to integrate (2), then we should have and in (3) instead of or.
so, for what reason do we end up with and in (1.1) but or in (3)?
to make clear my point , let's consider
$\int \frac{dy}{y}=\int \frac{dx}{x}$-----(4)
$=>\left\{\begin{array}{ll} \left\{\begin{array}{ll}ln|y|=ln|x|+c_1;y>0,x>0\\ \mathbf{or} \\ ln|y|=ln|x|+c_2;y<0,x>0 \end{array}\right. \\ \mathbf{and} \\ \left\{\begin{array}{ll}ln|y|=ln|x|+c_3;y>0,x<0\\ \mathbf{or} \\ln|y|=ln|x|+c_4;y<0,x<0\end{array}\right. \end{array}\right.$
see! , when it is the independent variable we use and(we say it's the same function) , but when its the dependent variable we use or( we say there are actually two different solutions, one of which will be eliminated by the initial condition). why do we do this ?
Whenever you encounter a problem in your equation, the typical fix is to find a solution for a specific portion without the problem. When you do that for two different portions of the dependent variable, you wind up with two different solutions that suggest different outputs for the same input, so they can’t both be the answer. When you do that to the input, then you’re defining solutions for two different sections so the different solutions can both be part of the same function.