If m and n are respectively order and degree of the differential equation (d^2y/dx^2)^5 + 6*((d^2y/dx^2)^3)/(d^3y/dx^3) + d^3y/dx^3 = x^2+5 then : Option 1: m = 3, n = 3 Option 2: m = 2, n = 3 Option 3: m = 3, n = 2 Option 4: m = 3, n = 5

Answer: m = 3, n = 2. Solution: Multiplying throughout by d^3y/dx^3 clears the fraction and produces a term (d^3y/dx^3)^2. The highest derivative present is of order 3, so m=3, and its highest power is 2, so n=2.

Calculus

Differential Equations

Medium

common

If $m$ and $n$ are respectively order and degree of the differential equation

$\left(\dfrac{d^2y}{dx^2}\right)^5 + 6\cdot\dfrac{\left(\dfrac{d^2y}{dx^2}\right)^3}{\dfrac{d^3y}{dx^3}} + \dfrac{d^3y}{dx^3} = x^2+5$ then :

m = 3, n = 3

m = 2, n = 3

m = 3, n = 2

m = 3, n = 5

✅ Correct Option: 3

Multiplying throughout by $\frac{d^3y}{dx^3}$ clears the fraction and produces a term $\left(\frac{d^3y}{dx^3}\right)^2$ . The highest derivative present is of order $3$ , so $m=3$ , and its highest power is $2$ , so $n=2$ .

2025: 22 May Shift 2

Match List-I with List-II

List-I	List-II
Differential Equation	Order and Degree
(A) $\left(\frac{d^2y}{dx^2}\right)^2 = e^x\left(\frac{dy}{dx}\right)^4 + 1 = 0$	(I) order = 1 and degree = 2
(B) $\left(\frac{dy}{dx}\right)^2 + xy = 0$	(II) order = 2 and degree = 1
(C) $\left(1 + \frac{dy}{dx}\right)^{3/2} = 4\left(\frac{d^2y}{dx^2}\right)^2$	(III) order = 2 and degree = 2
(D) $\sqrt\frac{d^2y}{dx^2} + 1 = \frac{dy}{dx}$	(IV) order = 2 and degree = 4

Choose the correct answer from the options given below:

2026: 19 May Shift 1

The solution of the initial value problem $x\frac{dy}{dx} = 2, y(1) = 2$ is

2026: 22 May Shift 1

General solution of the differential equation $\frac{dy}{dx}=e^{\frac{x^2}{2}}+xy$ is: (where C is an arbitrary constant)

2025: 3 June Shift 1

Match List-I with List-II

List-I	List-II
Differential Equation	Sum of order and degree
(A) $\frac{d^2y}{dx^2} + \frac{dy}{dx} + 3y = \sin x$	(I) 2
(B) $\frac{dy}{dx} = \sin(x + y)$	(II) 3
(C) $\sqrt{1 + (\frac{dy}{dx})^2} = \frac{d^2y}{dx^2}$	(III) 4
(D) $x^2(\frac{d^2y}{dx^2})^3 + y(\frac{dy}{dx})^4 + y^5 = 0$	(IV) 5

Choose the correct answer from the options given below:

2025: 3 June Shift 1

If m and n are respectively the order and degree of the differential equation $(\frac{d^2y}{dx^2})^{2} + (\frac{dy}{dx})^3 + y= 4x$ , then the value of $m + n$ is:

2026: 19 May Shift 1

Match List-I with List-II

List-I	List-II
(A) $y + \frac{dy}{dx} = \frac{1}{4}\int y\,dx$	(I) Order 1, Degree 1
(B) $y = \frac{dy}{dx} + \frac{C}{dy/dx}$	(II) Order 2, Degree 2
(C) $(xy^2+x)dx + (y-x^2)dy = 0$	(III) Order 2, Degree 1
(D) $\left[\left\{1+\left(\frac{dy}{dx}\right)^2\right\}^{3/2}/\frac{d^2y}{dx^2}\right] = K$	(IV) Order 1, Degree 2

Choose the correct answer from the options given below:

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