¿Por qué no podemos aplicar el Método de separación de variables a ecuaciones Diff de segundo orden?

Question

Supongamos que estamos resolviendo el siguiente 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

Tienes que ser precavido: recuerda que $$\frac{d^2y}{dx^2} = \frac{d\left(\frac{dy}{dx}\right)}{dx}$$ 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

Porque no puede escribir $$\dfrac{d^2 y}{y}=\dfrac{dx^2}{x}$$ 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 $\ int \ dfrac {dx ^ 2} {x}$ son ambos ilegales.