The integrating factor of the differential equation dy/dx = x + xy is Option 1: e^(-x/2) Option 2: e^(-x^2/2) Option 3: e^(x^2/2) Option 4: e^(x/2)

Answer: e^(-x^2/2). Solution: The differential equation is dy/dx = x + xy. To find the integrating factor, the equation must be written in standard form: dy/dx + P(x) * y = Q(x) dy/dx = x + xy dy/dx = x(1 + y) dy/dx - xy = x The equation is now in standard form with P(x) = -x and Q(x) = x. The integrating factor is given by: I.F. = e^( P(x)dx) I.F. = e^( (-x)dx) (-x)dx = -x^2/2 Therefore, the integrating factor is: I.F. = e^(-x^2/2)

Calculus

Differential Equations

Medium

core

The integrating factor of the differential equation $\frac{dy}{dx} = x + xy$ is

$e^{-x/2}$

$e^{-x^2/2}$

$e^{x^2/2}$

$e^{x/2}$

✅ Correct Option: 2

The differential equation is $\frac{dy}{dx} = x + xy$ .

To find the integrating factor, the equation must be written in standard form:

$\frac{dy}{dx} + P(x) \cdot y = Q(x)$

$\frac{dy}{dx} = x + xy$

$\frac{dy}{dx} = x(1 + y)$

$\frac{dy}{dx} - xy = x$

The equation is now in standard form with $P(x) = -x$ and $Q(x) = x$ .

The integrating factor is given by:

$\text{I.F.} = e^{\int P(x)dx}$

$\text{I.F.} = e^{\int (-x)dx}$

$\int (-x)dx = -\frac{x^2}{2}$

Therefore, the integrating factor is:

$\text{I.F.} = e^{-x^2/2}$

2025: 16 May Shift 1

Given differential equation, (1 + y²)dx = (tan⁻¹y - x)dy, then which of the following is/are true?

(A) Integrating factor = tan⁻¹x

(B) Integrating factor = tan⁻¹y

(D) Degree = 1

Choose the correct answer from the options given below:

2025: 14 May Shift 1

Which of the following are linear first order differential equations?

(A) $\frac{dy}{dx} + P(x)y = Q(x)$

(B) $\frac{dx}{dy} + P(y)x = Q(y)$

(D) $(1 + x^2)\frac{dy}{dx} + 2xy = 2$

Choose the correct answer from the options given below:

2026: 30 May Shift 2

If $y = x\frac{dy}{dx} + \sqrt{a^2\left(\frac{dy}{dx}\right)^2 + b^2}$ , then the order and degree of the differential equation are

2025: 26 May Shift 2

The particular solution of the differential equation $\frac{dy}{dx} = e^{x^2/2} + xy$ , when $x = 0$ , $y = 1$ , is

2025: 19 May Shift 1

Match List-I with List-II

List-I	List-II
Differential Equation	Order and degree of differential equation
(A) $\frac{d^2y}{dx^2} + 2\left(\frac{dy}{dx}\right)^2 = e^{\frac{dy}{dx}} + 1$	(I) Order = 1, Degree = 2
(B) $\left(\frac{d^2y}{dx^2}\right)^2 + 4\left(\frac{dy}{dx}\right)^3 = e^y - 1$	(II) Order = 2, Degree = 1
(C) $3\left(\frac{dy}{dx}\right) + 4y + e^y = \frac{dx}{dy}$	(III) Order = 2, Degree = 2
(D) $\frac{d^2y}{dx^2} + 3\left(\frac{dy}{dx}\right) = \left(e^y + \frac{dy}{dx}\right)^2$	(IV) Order = 2, Degree = Not defined

Choose the correct answer from the options given below:

2025: 13 May Shift 1

The general solution of the differential equation $ydx - (x + 2y^2)dy = 0$

The integrating factor of the differential equation dydx=x+xy\frac{dy}{dx} = x + xydxdy​=x+xy is

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The integrating factor of the differential equation $\frac{dy}{dx} = x + xy$ is