What is meant by "......The given differential equation is not polynomial equation in its derivatives and so it's degree is not defined"

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Solution

The degree of a differential equation is the power of the highest order derivative in the equation. . So basically to be able to get the degree of a differential equation, we need only the derivatives and their exponents to be part of the differential equation, that is like a polynomial in derivatives. The degree will not be defined if the highest order derivative is not raised to an exponent but is part of some other function for example:
log(y'') + y' +y = sin x . Here y'' is part of log function so no degree can be assigned to it.

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\frac{d^{2} y}{d x^{2}} + {(\frac{d y}{d x})}^{2} + sin (\frac{d y}{d x}) + 1 = 0

\frac{d^{2} y}{d x^{2}} + c o s (\frac{d y}{d x}) = 0 :

{(\frac{d^{2} y}{d x^{2}})}^{3} + {(\frac{d y}{d x})}^{2} + \sin (\frac{d y}{d x}) + 1 = 0

{(\frac{d^{2} y}{d x^{2}})}^{3} + e^{\frac{d y}{d x}} = 0

(a) 1 (b) 2

(c) 3 (d) not defined

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