Q.The order and degree of the differential equation $\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^{1/4}+x^{1/5}=0$, respectively, are:
(A) 2 and not defined
(B) 2 and 2
(C) 2 and 3
(D) 3 and 3

Order is the highest derivative present: here that is $\frac{d^2y}{dx^2}$, so the order is $2$.

Degree is defined only when the equation is a polynomial in its derivatives. The term $\left(\frac{dy}{dx}\right)^{1/4}$ carries a fractional power of a derivative, so the equation is not polynomial in the derivatives and the degree is * …

The equation is

Order

Order = the order of the highest derivative appearing. The highest here is the second derivative $\frac{d^2y}{dx^2}$, so the order is 2.

Degree …

Method: Reading Order and Degree Together

Order and degree are answered in sequence: the order comes straight from the highest derivative; the degree needs the polynomial-in-derivatives test.

Steps

Step 1: Find the order.

Locate the highest-order derivative present — its order is the order of the equation, regardless of powers or other terms.

Step 2: Test whether a degree exists.

Check every derivative for a fractional power or a transcendental wrapper. A term like $\left(\frac{dy}{dx}\right)^{1/4}$ makes the equation non-polynomial in derivatives, so the degree is not defined. …

Common Mistakes

Mistake 1: Assigning a numerical degree despite the $\left(\frac{dy}{dx}\right)^{1/4}$ term.

Why it's wrong: a fractional power of a derivative means the equation is not polynomial in its derivatives, so the degree is not defined. Correct approach: spot the fractional exponent and stop — degree not defined.

Mistake 2: Letting the $x^{1/5}$ term worry you. …