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:


Above differential equation has its degree undefined. Differentiating it with respect to $x$


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.

  • $\begingroup$ How exactly is "polynomial form" and "degree" defined for this task? If this is bound only to the highest-order derivative, you do not need to do anything to find that the degree is 1, the original equation is already in polynomial form. $\endgroup$ May 6 at 8:46
  • $\begingroup$ $y''=ln(y')$ is not polynomial differential equation in $y'$ so degree can't be told 1. Reference toppr.com/guides/maths/differential-equations/… $\endgroup$
    – Jay
    May 6 at 9:28
  • $\begingroup$ From your link "The coefficient of any term containing the highest order derivative should just be a function of x, y, or some lower order derivative." Note that $y'$ is a "lower order derivative". $\endgroup$ May 6 at 9:37
  • $\begingroup$ $ln$ is not coefficient of $y'$ and $y' $can be coeff of $y"$ is statement meaning 1st point in link for condition of order and degree identification :All of the derivatives in the equation are free from fractional powers, positive as well as negative if any. Am I correct? $\endgroup$
    – Jay
    May 6 at 9:40

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.

  • $\begingroup$ The degree of my original question is clearly stated undefined as per my sir. And $ln(y')$ is not a polynomial function in $y'$ $\endgroup$
    – Jay
    May 6 at 9:56
  • $\begingroup$ Then you use a definition that is different from the toppr link. $\endgroup$ May 6 at 9:58

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