Q.The degree of the differential equation $\left[1+\left(\frac{dy}{dx}\right)^2\right]^{3/2}=\frac{d^2y}{dx^2}$ is:
(A) 4
(B) $\frac{3}{2}$
(C) not defined
(D) 2

Concept: Degree of a differential equation is defined only when the equation is a polynomial in the derivatives. The degree is the power of the highest-order derivative after removing radicals and fractions.

Step 1: The given equation is

Step 2: To express it as a polynomial in derivatives, square both sides:

$\left[1+\left(\frac{dy}{dx}\right)^2\right]^{3} = \left(\frac{d^2y}{dx^2}\right)^2.$ …

The degree of a differential equation is the power of the highest-order derivative, provided the equation is a polynomial in all the derivatives that appear. If the equation involves fractional powers, roots, or non-polynomial functions of the derivatives, the degree is not defined until we rewrite it as a polynomial — if possible.

In this problem, the highest-order derivative is $\frac{d^2y}{dx^2}$, and it appears inside a square root (via the exponent $\frac{3}{2}$ on the left side). That fractional exponent is the obstacle: the equation is not yet a polynomial in the derivatives. So we must first remove the fractional power by raising both sides to a suitable integer power, then check the polynomial form.

Let’s work through it step by step.

1. Identify the highest-order derivative. The equation is

The highest derivative present is $\frac{d^2y}{dx^2}$ (order 2). The degree concerns this derivative, but only after the equation is made polynomial in all derivatives.

2. Remove the fractional exponent. The left side has exponent $3/2$. To clear the square root, square both sides:

This gives

3. Check if the equation is now a polynomial in derivatives. …

Method: Degree After Clearing a Fractional Power

When a derivative sits under a fractional exponent, you cannot read the degree directly — first rationalise the equation into a polynomial in the derivatives, then read the power of the highest-order derivative.

Steps

Step 1: Isolate the radical/fractional-power term.

Get the term carrying the fractional exponent (e.g. a $\tfrac{3}{2}$ power) alone on one side.

Step 2: Raise both sides to clear the fraction.

Raise to whatever integer power removes the denominator in the exponent — a $\tfrac{3}{2}$ power is cleared by squaring:

$\left[1+\left(\frac{dy}{dx}\right)^2\right]^{3}=\left(\frac{d^2y}{dx^2}\right)^2 .$ …

Common Mistakes

Mistake 1: Reading the degree as $\tfrac{3}{2}$ straight from the exponent.

Why it's wrong: degree is defined only for a polynomial in the derivatives, so the fractional power must be cleared first. Correct approach: square both sides, then read the power of the highest-order derivative.

Mistake 2: Squaring only one side or mishandling $\left[\,\cdot\,\right]^{3/2}$. …