The differential equation representing family of curves y = e^(-2x)(a x + b x), where a and b are arbitrary constant, is : Option 1: (d^2 y)/dx^2 + y^2 = 0 Option 2: (d^2 y)/dx^2 + 4 dy/dx + 5y = 0 Option 3: (d^2 y)/dx^2 - 4 dy/dx - 5y = 0 Option 4: (d^2 y)/dx^2 - y^2 = 0

Answer: (d^2 y)/dx^2 + 4 dy/dx + 5y = 0. Solution: The form y = e^(-2x)(a x + b x) corresponds to roots -2 ± i of the auxiliary equation. Auxiliary equation: (m - (-2+i))(m - (-2-i)) = (m+2)^2 + 1 = m^2 + 4m + 5 = 0. Differential equation: d^2y/dx^2 + 4dy/dx + 5y = 0.

Calculus

Differential Equations

Medium

The differential equation representing family of curves $y = e^{-2x}(a \cos x + b \sin x)$ , where a and b are arbitrary constant, is :

$\frac{d^2 y}{dx^2} + y^2 = 0$

$\frac{d^2 y}{dx^2} + 4 \frac{dy}{dx} + 5y = 0$

$\frac{d^2 y}{dx^2} - 4 \frac{dy}{dx} - 5y = 0$

$\frac{d^2 y}{dx^2} - y^2 = 0$

✅ Correct Option: 2

The form $y = e^{-2x}(a \cos x + b \sin x)$ corresponds to roots $-2 \pm i$ of the auxiliary equation.

Auxiliary equation: $(m - (-2+i))(m - (-2-i)) = (m+2)^2 + 1 = m^2 + 4m + 5 = 0$ .

Differential equation: $\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 5y = 0$ .

2026: 22 May Shift 1

Integrating factor of the differential equation $x(x-1)\frac{dy}{dx}-(x-2)y=x^3(2x-1), x>1$ is:

2026: 21 May Shift 2

Let $y = f(x)$ be the solution curve of the differential equation $\dfrac{dy}{dx} + 2y^2 = 0$ if $f(1) = 1$ , then $f(2)$ is equal to

2026: 19 May Shift 1

The general solution of the given differential equation $\frac{dy}{dx} = \frac{3e^{2x}+3e^{4x}}{e^{x}+e^{-x}}$ (where C is an arbitrary constant)

2025: 26 May Shift 2

Match List-I with List-II

List-I	List-II
Differential Equations	Order and degree
(A) $ydx + x\log(y/x)dy - 2xdy = 0$	(I) Order : 2, degree:1
(B) $\left(\frac{d^3y}{dx^3}\right)^2 + 3\frac{d^2y}{dx^2} + 2\left(\frac{dy}{dx}\right)^4 = y^2$	(II) Order :1, degree:1
(C) $\frac{dy}{dx} + \log\left(\frac{dy}{dx}\right) + x = y$	(III) Order : 3, degree:2
(D) $\left(\frac{ds}{dt}\right)^4 + 2s\frac{d^2s}{dt^2} = 0$	(IV) Order : 1, degree: Not defined

Choose the correct answer from the options given below:

2022: 25 May Shift 1

The degree of the differential equation

$\left(1 + \frac{dy}{dx}\right)^4 = \left(\frac{d^2y}{dx^2}\right)^2$ is:

2026: 25 May Shift 1

General solution of the differential equation $(e^x + e^{-x})dy = (3e^{2x} + 3e^{4x})dx$ is:

[Where C is an arbitrary constant]

The differential equation representing family of curves y=e−2x(acos⁡x+bsin⁡x)y = e^{-2x}(a \cos x + b \sin x)y=e−2x(acosx+bsinx), where a and b are arbitrary constant, is :

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CUET Mathematics Past Year Questions (Topic-Wise)

The differential equation representing family of curves $y = e^{-2x}(a \cos x + b \sin x)$ , where a and b are arbitrary constant, is :