The order and degree of the differential equation $\left[1+\left(\frac{dy}{dx}\right)^2\right]=\frac{d^2y}{dx^2}$ are: (A) $2,\ \frac{3}{2}$ (B) 2, 3 (C) 2, 1 (D) 3, 4

> [!TLDR] > The order is the highest derivative present (2), and the degree is the power of that highest derivative after removing radicals and fractions — here the equation is already polynomial in $\frac{d^2y}{dx^2}$ with exponent 1, so degree is 1. The correct option is **(C)**. The order of a differential equation is simply the highest derivative that appears. The degree is trickier: it’s the power of the highest derivative *after* the equation has been made polynomial in all derivatives — meaning you must clear any radicals, fractions, or fractional powers that involve the derivatives. Here the equation is: $$ 1 + \left(\frac{dy}{dx}\right)^2 = \frac{d^2y}{dx^2} $$ Let’s walk through it. 1. **Identify the highest derivative.** The left side has $\frac{dy}{dx}$ (first derivative) and a constant. The right side has $\frac{d^2y}{dx^2}$ (second derivative). No derivative higher than the second appears, so the **order is 2**. 2. **Check if the equation is already polynomial in the derivatives.** The term $\left(\frac{dy}{dx}\right)^2$ is a polynomial term (power 2). The term $\frac{d^2y}{dx^2}$ appears with exponent 1. There are no square roots, no fractional powers, no denominators containing derivatives. The equation is already in the form: $$ \frac{d^2y}{dx^2} - \left(\frac{dy}{dx}\right)^2 - 1 = 0 $$ which is a polynomial in $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$. 3. **Read off the degree.** …

The order and degree of the differential equation [1+((dy)/(dx))²]=(d²y)/(dx²) are: (A) 2,(3)/(2) (B) 2, 3 (C)

NCERT Exemplar · Q90

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

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The order is the highest derivative present (2), and the degree is the power of that highest derivative after removing radicals and fractions — here the equation is already polynomial in $\frac{d^2y}{dx^2}$ with exponent 1, so degree is 1. The correct option is (C).

The order of a differential equation is simply the highest derivative that appears. The degree is trickier: it’s the power of the highest derivative after the equation has been made polynomial in all derivatives — meaning you must clear any radicals, fractions, or fractional powers that involve the derivatives.

Here the equation is:

1 + \left(\frac{dy}{dx}\right)^2 = \frac{d^2y}{dx^2}

Let’s walk through it.

Identify the highest derivative.

The left side has $\frac{dy}{dx}$ (first derivative) and a constant. The right side has $\frac{d^2y}{dx^2}$ (second derivative). No derivative higher than the second appears, so the order is 2.
Check if the equation is already polynomial in the derivatives.

The term $\left(\frac{dy}{dx}\right)^2$ is a polynomial term (power 2). The term $\frac{d^2y}{dx^2}$ appears with exponent 1. There are no square roots, no fractional powers, no denominators containing derivatives. The equation is already in the form:

\frac{d^2y}{dx^2} - \left(\frac{dy}{dx}\right)^2 - 1 = 0

which is a polynomial in $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ .

Read off the degree. …

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Q.The order and degree of the differential equation $\left[1+\left(\frac{dy}{dx}\right)^2\right]=\frac{d^2y}{dx^2}$ are:
(A) $2,\ \frac{3}{2}$
(B) 2, 3
(C) 2, 1
(D) 3, 4

Concept understanding — Order Of Differential Equation

Order of a Differential Equation

Some examples

Why order matters …

Method: Order and degree when a lower derivative is raised to a power

Steps

Common Mistakes

Q.The order and degree of the differential equation [1+(dydx)2]=d2ydx2\left[1+\left(\frac{dy}{dx}\right)^2\right]=\frac{d^2y}{dx^2}[1+(dxdy​)2]=dx2d2y​ are: (A) 2, 322,\ \frac{3}{2}2, 23​ (B) 2, 3 (C) 2, 1 (D) 3, 4

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Q.The order and degree of the differential equation $\left[1+\left(\frac{dy}{dx}\right)^2\right]=\frac{d^2y}{dx^2}$ are:
(A) $2,\ \frac{3}{2}$
(B) 2, 3
(C) 2, 1
(D) 3, 4

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