In order to find a DE from the general solution of a first order equation, we use implicit differention, so should we try implicit differentition twice to find the DE from solution to a 2nd order equation?

How about

$\displaystyle y = c_{1}e^{-2x} + c_{2}e^{3x}$ ?

Now if we had a general solution to a first order one like

$\displaystyle y = x^{2} + C$

then

$\displaystyle 1(y') = 2x + 0$

$\displaystyle y' = 2x$

or

$\displaystyle \dfrac{dy}{dx} = 2x$

or

$\displaystyle dy = 2x dx$ would be the DE.