If c is an arbitrary constant, then Match the LIST-I with LIST-II, LIST-I (Differential Equation), LIST-II (Solution), A., dy/dx = 1 + x + y + xy, I., _e y = 2x + _e (x-1)^2 + c, B., (x-1)dy/dx = 2xy, II., y = cx, C., dy/dx = 1 - x + y - xy, III., _e 1+y = x + x^2/2 + c, D., x dy = y dx, IV., _e 1+y = x - x^2/2 + c, Choose the correct answer from the options given below: Option 1: A-III, B-I, C-IV, D-II Option 2: A-IV, B-I, C-III, D-II Option 3: A-III, B-IV, C-I, D-II Option 4: A-IV, B-III, C-II, D-I

Answer: A-III, B-I, C-IV, D-II. Solution: A: factorise as (1+x)(1+y), giving _e, 1+y, =x+x^2/2+c (III). B: dy/y=2x/(x-1)dx=2(1+1/(x-1))dx, giving _e, y, =2x+_e(x-1)^2+c (I). C: factorise as (1-x)(1+y), giving IV. D: dy/y=dx/x gives y=cx (II).

Calculus

Differential Equations

Medium

common

If $c$ is an arbitrary constant, then

Match the LIST-I with LIST-II

LIST-I (Differential Equation) LIST-II (Solution)
A. $\dfrac{dy}{dx} = 1 + x + y + xy$ I. $\log_e \vert y \vert = 2x + \log_e (x-1)^2 + c$
B. $(x-1)\dfrac{dy}{dx} = 2xy$ II. $y = cx$
C. $\dfrac{dy}{dx} = 1 - x + y - xy$ III. $\log_e \vert 1+y \vert = x + \dfrac{x^2}{2} + c$
D. $x\,dy = y\,dx$ IV. $\log_e \vert 1+y \vert = x - \dfrac{x^2}{2} + c$

Choose the correct answer from the options given below:

	LIST-I (Differential Equation)		LIST-II (Solution)
A.	$\dfrac{dy}{dx} = 1 + x + y + xy$	I.	$\log_e \vert y \vert = 2x + \log_e (x-1)^2 + c$
B.	$(x-1)\dfrac{dy}{dx} = 2xy$	II.	$y = cx$
C.	$\dfrac{dy}{dx} = 1 - x + y - xy$	III.	$\log_e \vert 1+y \vert = x + \dfrac{x^2}{2} + c$
D.	$x\,dy = y\,dx$	IV.	$\log_e \vert 1+y \vert = x - \dfrac{x^2}{2} + c$

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

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

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

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

✅ Correct Option: 1

A: factorise as $(1+x)(1+y)$ , giving $\log_e|1+y|=x+\frac{x^2}{2}+c$ (III). B: $\frac{dy}{y}=\frac{2x}{x-1}dx=2\left(1+\frac{1}{x-1}\right)dx$ , giving $\log_e|y|=2x+\log_e(x-1)^2+c$ (I). C: factorise as $(1-x)(1+y)$ , giving IV. D: $\frac{dy}{y}=\frac{dx}{x}$ gives $y=cx$ (II).

2025: 22 May Shift 2

The particular solution of the differential equation $\left[x \sin^2\left(\frac{y}{x}\right) - y\right]dx + xdy = 0$ , $y = \frac{\pi}{4}$ when $x = 1$ is

2022: 6 Aug Shift 2

The curve passing through the point $(-1, 1)$ , given that the slope of the tangent to the curve at any point $(x, y)$ is $\frac{2x}{y^2}$ also passes through the point $\left( k, -\frac{1}{2} \right)$ , then

2026: 14 May Shift 2

Match the LIST-I with LIST-II

LIST-I		LIST-II
A.	The degree of the differential equation $\left(\frac{d^3y}{dx^3}\right)^4+\left(\frac{d^2y}{dx^2}\right)^5+\left(\frac{dy}{dx}\right)+y=0$	I.	0
B.	The order of the differential equation $\left(\frac{d^3y}{dx^3}\right)^4+\left(\frac{d^2y}{dx^2}\right)^5+\left(\frac{dy}{dx}\right)+y^2=0$	II.	4
C.	The number of arbitrary constants in the general solution of a differential equation of second order is	III.	3
D.	The number of arbitrary constants in a particular solution of a differential equation of third order is	IV.	2

Choose the correct answer from the options given below:

2026: 21 May Shift 2

The general solution of the differential equation $(y-x)dy + (x+y)dx = 0$ is: (where C is arbitrary constant)

2025: 13 May Shift 2

The integrating factor of the differential equation, $x^2 \frac{dy}{dx} + xy = log_e x$ is equal to

2026: 22 May Shift 1

The particular solution of the differential equation $\log\left(\frac{dy}{dx}\right)=3x+4y$ , given that $y=0$ , when $x=0$ is:

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