Calculus

Differential Equations

Medium

The differential equation representing family of curves $y = ae^{mx} + be^{nx}$ , where $a$ and $b$ are arbitrary constants, is

$\frac{d^2y}{dx^2} + (m+n)\frac{dy}{dx} + y = 0$

$\frac{d^2y}{dx^2} + mn\frac{dy}{dx} + (m+n)y = 0$

$\frac{d^2y}{dx^2} - (m+n)\frac{dy}{dx} + mny = 0$

$\frac{d^2y}{dx^2} + (m+n)\frac{dy}{dx} - mny = 0$

✅ Correct Option: 3

$y = ae^{mx} + be^{nx}$ , so $y' = ame^{mx} + bne^{nx}$ and $y'' = am^2 e^{mx} + bn^2 e^{nx}$ . Verify: $y'' - (m+n) y' + mn\, y = am^2 e^{mx} + bn^2 e^{nx} - (m+n)(am e^{mx} + bn e^{nx}) + mn(a e^{mx} + b e^{nx}) = 0$ . So the differential equation is $y'' - (m+n)y' + mn\, y = 0$ .

2025: 3 June Shift 1

Match List-I with List-II

List-I	List-II
Differential Equation	Sum of order and degree
(A) $\frac{d^2y}{dx^2} + \frac{dy}{dx} + 3y = \sin x$	(I) 2
(B) $\frac{dy}{dx} = \sin(x + y)$	(II) 3
(C) $\sqrt{1 + (\frac{dy}{dx})^2} = \frac{d^2y}{dx^2}$	(III) 4
(D) $x^2(\frac{d^2y}{dx^2})^3 + y(\frac{dy}{dx})^4 + y^5 = 0$	(IV) 5

Choose the correct answer from the options given below:

2026: 15 May Shift 1

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.	$\log_e \vert 1+y \vert = x - \dfrac{x^2}{2} + c$

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2026: 18 May Shift 1

Which of the following differential equations are of order 2 and degree 1?

(A). $\frac{d}{dx}\left(\frac{dy}{dx}\right) + x\frac{dy}{dx} = y$

(B). $\frac{d^2y}{dx^2} + x = \sqrt{\frac{dy}{dx} - y}$

(C). $\left(\frac{dy}{dx}\right)^2 + \frac{d^2y}{dx^2} = \log x$

(D). $\left(\frac{dy}{dx}\right)^2 + \frac{dy}{{dx}} = 3$

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2022: 23 Aug Shift 1

The integrating factor of the differential equation $x\frac{dy}{dx} - y = 2x^2$ is :

2026: 19 May Shift 1

Match List-I with List-II

List-I	List-II
(A) $y + \frac{dy}{dx} = \frac{1}{4}\int y\,dx$	(I) Order 1, Degree 1
(B) $y = \frac{dy}{dx} + \frac{C}{dy/dx}$	(II) Order 2, Degree 2
(C) $(xy^2+x)dx + (y-x^2)dy = 0$	(III) Order 2, Degree 1
(D) $\left[\left\{1+\left(\frac{dy}{dx}\right)^2\right\}^{3/2}/\frac{d^2y}{dx^2}\right] = K$	(IV) Order 1, Degree 2

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2025: 30 May Shift 1

The solution of the differential equation $\log_e\left(\frac{dy}{dx}\right) = 5x + 2y$ is given by

The differential equation representing family of curves y=aemx+benxy = ae^{mx} + be^{nx}y=aemx+benx, where aaa and bbb are arbitrary constants, is

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The differential equation representing family of curves $y = ae^{mx} + be^{nx}$ , where $a$ and $b$ are arbitrary constants, is