Calculus

Differential Equations

Medium

common

Match List-I with List-II

List-I (Differential Equation) List-II (Order and degree)
(A) $\left(\dfrac{d^2y}{dx^2}\right)^2 + e^{\frac{dy}{dx}} = 0$ (I) order = 2, degree = 1
(B) $xy\dfrac{d^2y}{dx^2} + x\left(\dfrac{dy}{dx}\right)^2 - y\dfrac{dy}{dx} = 0$ (II) order = 2, degree = not defined
(C) $\left(\dfrac{d^3y}{dx^3}\right)^2 + \log_e\left(\dfrac{dy}{dx}\right) = 4x$ (III) order = 3, degree = 2
(D) $\dfrac{d}{dx}\left(\dfrac{d^2y}{dx^2}\right) + y\dfrac{dy}{dx} = \left(1+\dfrac{dy}{dx}\right)^{1/2}$ (IV) order = 3, degree = not defined

Choose the correct answer from the options given below:

List-I (Differential Equation)	List-II (Order and degree)
(A) $\left(\dfrac{d^2y}{dx^2}\right)^2 + e^{\frac{dy}{dx}} = 0$	(I) order = 2, degree = 1
(B) $xy\dfrac{d^2y}{dx^2} + x\left(\dfrac{dy}{dx}\right)^2 - y\dfrac{dy}{dx} = 0$	(II) order = 2, degree = not defined
(C) $\left(\dfrac{d^3y}{dx^3}\right)^2 + \log_e\left(\dfrac{dy}{dx}\right) = 4x$	(III) order = 3, degree = 2
(D) $\dfrac{d}{dx}\left(\dfrac{d^2y}{dx^2}\right) + y\dfrac{dy}{dx} = \left(1+\dfrac{dy}{dx}\right)^{1/2}$	(IV) order = 3, degree = not defined

(A) - (I), (B) - (III), (C) - (IV), (D) - (II)

(A) - (III), (B) - (IV), (C) - (II), (D) - (I)

(A) - (II), (B) - (I), (C) - (IV), (D) - (III)

(A) - (IV), (B) - (III), (C) - (II), (D) - (I)

✅ Correct Option: 3

Degree exists only when the equation is polynomial in all derivatives.

(A) order 2, but $e^{y'}$ makes degree undefined. (B) order 2 and polynomial with $y''$ appearing to power 1, so degree 1. (C) order 3 with $\log_e(y')$ , so degree undefined. (D) is $y'''+yy'=(1+y')^{1/2}$ ; squaring makes it polynomial of order 3 and degree 2.

2025: 15 May Shift 1

The general solution of the differential equation $(1 + e^x)dy + ye^x dx = 0$ , where $y > 0$ , is

2022: 4 Aug Shift 1

Match List I with List II

List I	List II : Order and degree respectively
A. $\frac{dy}{dx} - (x^2+3) = 0$	I. 2 and 1
B. $2x^2 \frac{d^2y}{dx^2} - 3\frac{dy}{dx} + y = 0$	II. 2 and 3
C. $y''' + y^2 + e^{y'} = 0$	III. 1 and 1
D. $\left(\frac{ds}{dt}\right)^4 + 3s\left(\frac{d^2s}{dt^2}\right)^3 = 0$	IV. 3 and not defined

Choose the correct answer from the options given below:

2025: 15 May Shift 2

Match List-I with List-II

List-I	List-II
Differential equation	Order and degree
(A) $(y'')^3 + (y')^4 - 6 = (y''')^2$	(I) Order = 1, Degree = 2
(B) $\sqrt{(y')^2 + 5} = y''$	(II) Order = 2, Degree = 3
(C) $(y')^2 = (2 + y'')^{3/2}$	(III) Order = 2, Degree = 2
(D) $y = xy' + \sqrt{a^2(y')^2 + b^2}$	(IV) Order = 3, Degree = 2

Choose the correct answer from the options given below:

2026: 30 May Shift 2

The differential equation representing family of circles with centre at origin and radius r units is-

2025: 13 May Shift 1

The particular solution of the differential equation $e^x\sqrt{1-y^2}dx + \frac{y}{x}dy = 0$ , given that $y = 1$ , when $x = 0$ is:

2025: 15 May Shift 2

Consider the curve which is represented by the differential equation $\frac{dy}{dx} = 1 + x + y + xy$ . If it passes through the point $(0,0)$ , then which of the following is/are true?

(A) it is a straight line.

(B) it is a parabola.

(D) Its equation is $xy(x + 1)\left(y - \frac{1}{\sqrt{e}} + 1\right) = 0$

Choose the correct answer from the options given below:

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