Suppose we are solving the following ODE $${\mathrm{d}y\over \mathrm{d}x}={y\over x}$$ then we can solve it by seperation of variables method. But if we have to solve $${\mathrm{d}^2y\over \mathrm{d}x^2} = {y\over x}$$ then why cant we seperate $\mathrm{d}^2y\over \mathrm{d}x^2$?
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$\begingroup$ Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. $\endgroup$ – José Carlos Santos Aug 20 '18 at 11:32
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You have to be cautious: remember that $$\frac{d^2y}{dx^2} = \frac{d\left(\frac{dy}{dx}\right)}{dx}$$ So you could use separation of variables, but your ODE will become $$\frac{1}{y}d\left(\frac{dy}{dx}\right) = \frac{1}{x}dx$$ and not $$\frac{1}{y}dydy = \frac{1}{x}dxdx\;\;\;\;\color{red}{\text{Wrong}}$$
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Because you can't write $$\dfrac{d^2 y}{y}=\dfrac{dx^2}{x}$$or equivalently $$\dfrac{d^2y}{y}=\dfrac{dx \cdot dx}{x}$$this is meaningless. Also you would be stopped in the next step. $\int \dfrac{d^2y}{y}$ and $\int\dfrac{dx^2}{x}$ are both illegal.
