General solution of the differential equation (1+x^2)dy + 2xy dx = x dx is: (where C is an arbitrary constant) Option 1: y(1+x^2) = _e, x, + c Option 2: x(1+y^2) = _e, x, + c Option 3: y(1+x^2) = _e, x, + c Option 4: x(1+y^2) = _e, x, + c

Answer: y(1+x^2) = _e, x, + c. Solution: The left side is an exact derivative: (1+x^2)dy + 2xy dx = d[y(1+x^2)]. So d/dx[y(1+x^2)] = x. Integrating gives y(1+x^2) = _e, x, + c.

Calculus

Differential Equations

Medium

core

General solution of the differential equation $(1+x^2)dy + 2xy\,dx = \cot x\,dx$ is:

(where $C$ is an arbitrary constant)

$y(1+x^2) = \log_e|\sin x| + c$

$x(1+y^2) = \log_e|\sin x| + c$

$y(1+x^2) = \log_e|\cos x| + c$

$x(1+y^2) = \log_e|\cos x| + c$

✅ Correct Option: 1

The left side is an exact derivative: $(1+x^2)dy + 2xy\,dx = d\left[y(1+x^2)\right]$ .

So $\dfrac{d}{dx}\left[y(1+x^2)\right] = \cot x$ .

Integrating gives $y(1+x^2) = \log_e|\sin x| + c$ .

2026: 18 May Shift 1

Match List-I with List-II

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

Choose the correct answer from the options given below:

2026: 21 May Shift 1

Match List-I with List-II

List-I (Differential Equation)	List-II (Degree)
(A) $2(y'')^2 + 3y = 6$	(I) 4
(B) $2y + 5(y')^4 = 7$	(II) 1
(C) $\log x + (y'')^3 = e^x$	(III) 2
(D) $\sqrt{1-2y'} = 5(y'')^{\frac{1}{4}}$	(IV) 3

Choose the correct answer from the options given below:

2026: 26 May Shift 2

If the order and degree of the differential equation $y + \dfrac{dy}{dx} = \int y\,dx$ are m and n respectively, then (m - n) is equal to

2023: 23 May Shift 3

The solution of $y' - y'' = 2x$ is:

A. $y = x^2 + 2x + 2$

B. $y = x^2 + 2x + 1$

C. $y = x + 2$

D. $y = x^2 - 2x + 1$

Choose the correct answer from the options given below:

2023: 15 June Shift 2

The sum of order and degree of the differential equation $\frac{\left\{1+\left(\frac{dy}{dx}\right)^2\right\}^{\frac{5}{2}}}{\frac{d^2y}{dx^2}} = p$ is :

2025: 21 May Shift 1

The general solution of the differential equation $x(1 + y^2)dx + y(1 + x^2)dy = 0$ is

General solution of the differential equation (1+x2)dy+2xy dx=cot⁡x dx(1+x^2)dy + 2xy\,dx = \cot x\,dx(1+x2)dy+2xydx=cotxdx is: (where CCC is an arbitrary constant)

Solution

Related questions:

Keyboard Shortcuts

CUET Mathematics Past Year Questions (Topic-Wise)

General solution of the differential equation $(1+x^2)dy + 2xy\,dx = \cot x\,dx$ is:

(where $C$ is an arbitrary constant)