Why can’t we apply Method of separation of variables to second order Diff equations?

Question

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 $\mathrm{d}^2y\over \mathrm{d}x^2$?

Answer 1

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 $$\frac{1}{y}dydy = \frac{1}{x}dxdx\;\;\;\;\color{red}{\text{Wrong}}$$

Answer 2

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.