Can we differentiate a differential equation to determine its order and degree? It is known that we must need to convert the differential equation in polynomial equation of differential coefficients. But Can we differentiate a differential equation (whose degree is not defined) to determine its order and degree ?
Example to show my doubt clearly:
$y''=e^{y'}$
Above differential equation has its degree undefined. Differentiating it with respect to $x$
$y'''=(y'')^2$
So we may conclude that this is third order differential equation with degree $=1$
I know that on differentiating a differential equation number of arbitrary constants of solution equation increases hence we should not differentiate a differential equation in general. But I did not found a reference which states that we cannot differentiate a differential equation to determine its order and degree so I want to confirm my thoughts.
According to the link provided, a second order ODE can be said to have a degree if it is of the form
$$
(y'')^d+a_{d-1}(x,y,y')(y'')^{d-1}+...+a_1(x,y,y')y''+a_0(x,y,y')=0
$$
where the coefficient functions are sufficiently regular. The degree is then $d$.
So in the equation in question we have $d=1$ and $a_0(x,y,y')=-\ln(y')$, falling into the required pattern.

Note that assigning a degree to an ODE, and declaring what ODE can be assigned a degree, is a matter of opinion and circumstance. For example, in the intersection of differential equation and algebraic geometry a different definition might be more fruitful.
According to the link provided, a second order ODE can be said to have a degree if it is of the form
$$
(y'')^d+a_{d-1}(x,y,y')(y'')^{d-1}+...+a_1(x,y,y')y''+a_0(x,y,y')=0
$$
where the coefficient functions are sufficiently regular. The degree is then $d$.
So in the equation in question we have $d=1$ and $a_0(x,y,y')=-\ln(y')$, falling into the required pattern.

Note that assigning a degree to an ODE, and declaring what ODE can be assigned a degree, is a matter of opinion and circumstance. For example, in the intersection of differential equation and algebraic geometry a different definition might be more fruitful.