The integrating factor of the differential equation y_e(10y).dx/dy+x=1/([_e(10y)]^2), is Option 1: _e y Option 2: _e(10y) Option 3: _e 10 Option 4: 1+_e y

Answer: _e(10y). Solution: Divide throughout by y_e(10y) to get the linear form dx/dy + x/(y_e(10y)) = Q(y). The integrating factor is e^( dy/(y_e(10y))). Putting u = _e(10y) gives du = dy/y, so the integral is _e u. Hence the integrating factor is _e(10y).

Calculus

Differential Equations

Medium

core

The integrating factor of the differential equation $y\log_e(10y).\dfrac{dx}{dy}+x=\dfrac{1}{[\log_e(10y)]^2}$ , is

$\log_e y$

$\log_e(10y)$

$\log_e 10$

$1+\log_e y$

✅ Correct Option: 2

Divide throughout by $y\log_e(10y)$ to get the linear form $\dfrac{dx}{dy} + \dfrac{x}{y\log_e(10y)} = Q(y)$ .

The integrating factor is $e^{\int \frac{dy}{y\log_e(10y)}}$ . Putting $u = \log_e(10y)$ gives $du = dy/y$ , so the integral is $\log_e u$ .

Hence the integrating factor is $\log_e(10y)$ .

2026: 26 May Shift 2

Match the LIST-I with LIST-II

	LIST-I Differential equation		LIST-II Order and Degree
A.	$\left(\dfrac{dy}{dx}\right)^2 + 3y\left(\dfrac{d^2y}{dx^2}\right) = 0$	I.	$1,1$
B.	$\sqrt{1-\left(\dfrac{dy}{dx}\right)^2} = a\left(\dfrac{d^2y}{dx^2}\right)^{1/3}$	II.	$1,2$
C.	$y = \dfrac{dy}{dx} + \dfrac{5}{(dy/dx)}$	III.	$2,1$
D.	$\sqrt{1-y^2}\,dx + \sqrt{1-x^2}\,dy = 0$	IV.	$2,2$

Choose the correct answer from the options given below:

2026: 25 May Shift 1

Which of the following is a homogeneous differential equation ?

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: 12 May Shift 2

The solution of the differential equation $\dfrac{dy}{dx} = x^2y^2$ with $y = 1$ at $x = 0$ is

2026: 25 May Shift 1

The product of order and degree of the differential equation

$x\dfrac{d^3y}{dx^3} = \left(1 + \left(\dfrac{dy}{dx}\right)^2\right)^4$ is

2025: 21 May Shift 2

Match List-I with List-II

List-I	List-II
(A) Integrating factor of $xdy - (y + x^2)dx = 0$	(I) $x^2$
(B) Integrating factor of $xdy + (2y + x^2)dx = 0$	(II) $x^3$
(C) Integrating factor of $(3y - x^2)dx + xdy = 0$	(III) $x$
(D) Integrating factor of $(y + 3x^2)dx + xdy = 0$	(IV) $\frac{1}{x}$

Choose the correct answer from the options given below:

The integrating factor of the differential equation ylog⁡e(10y).dxdy+x=1[log⁡e(10y)]2y\log_e(10y).\dfrac{dx}{dy}+x=\dfrac{1}{[\log_e(10y)]^2}yloge​(10y).dydx​+x=[loge​(10y)]21​, is

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The integrating factor of the differential equation $y\log_e(10y).\dfrac{dx}{dy}+x=\dfrac{1}{[\log_e(10y)]^2}$ , is