Integerating factor of (x _e x) dy/dx + y = 2 _e x is : Option 1: x Option 2: e^x Option 3: _e x Option 4: _e (_e x)

Answer: _e x. Solution: Rewrite in standard form: dy/dx+1/(x x)y=2/x. Integrating factor =e^(/(x x))=e^(( x))= x.

Calculus

Differential Equations

Medium

core

Integerating factor of $(x \log_e x) \frac{dy}{dx} + y = 2 \log_e x$ is :

$x$

$e^x$

$\log_e x$

$\log_e (\log_e x)$

✅ Correct Option: 3

Rewrite in standard form: $\frac{dy}{dx}+\frac{1}{x\log x}y=\frac{2}{x}$ .

Integrating factor $=e^{\int\frac{dx}{x\log x}}=e^{\log(\log x)}=\log x$ .

2025: 3 June Shift 2

The general solution of the differential equation $\frac{dy}{dx} = -4xy^2$ is given by

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: 29 May Shift 1

Which of the following are correct?

(A) The order of the differential equation $\left(\dfrac{dy}{dx}\right)^2+3y^2-5y=0$ is 2.

(B) The degree of the differential equation $x\dfrac{d^2y}{dx^2}=\left(3+2\left(\dfrac{dy}{dx}\right)^2\right)^3$ is 3.

(D) The general solution of an ordinary differential equation of $n^{th}$ order contains $n$ independent arbitrary constants.

Choose the correct answer from the options given below:

2026: 18 May Shift 1

Match List-I with List-II

Choose the correct answer from the options given below:

2026: 18 May Shift 2

The order and degree of the differential equation $\left[2+\left(\dfrac{dy}{dx}\right)\right]^{5/2}=7\dfrac{d^3y}{dx^3}$ is

2026: 15 May Shift 1

The solution of the differential equation $\dfrac{dy}{dx} = \dfrac{x^2+xy+y^2}{x^2}$ for $x>0, y>0$ at $x=1, y=2$ is