The equation of curve whose slope is given by dy/dx = x and which passes through (1, 5/2) is : Option 1: y = x^2 + 5/2 Option 2: y = x^2/2 + 2 Option 3: y = x^2/3 + 2/3 Option 4: y = 3x^2 + 5

Answer: y = x^2/2 + 2. Solution: Integrating dy/dx=x: y=x^2/2+C. Using (1, 5/2): 5/2=1/2+C, so C=2. Therefore y=x^2/2+2.

Calculus

Differential Equations

Easy

core

The equation of curve whose slope is given by $\frac{dy}{dx} = x$ and which passes through $\left(1, \frac{5}{2}\right)$ is :

$y = x^2 + \frac{5}{2}$

$y = \frac{x^2}{2} + 2$

$y = \frac{x^2}{3} + \frac{2}{3}$

$y = 3x^2 + 5$

✅ Correct Option: 2

Integrating $\frac{dy}{dx}=x$ : $y=\frac{x^2}{2}+C$ .

Using $(1, 5/2)$ : $5/2=1/2+C$ , so $C=2$ .

Therefore $y=\frac{x^2}{2}+2$ .

2025: 21 May Shift 1

The differential equation representing the curve $y = e^{2x}(a + bx)$ , where a, b are arbitrary constants is

2026: 19 May Shift 1

The general solution of the differential equation $\frac{dy}{dx} + y\cot x = 2\cos x$ is (Where C is an arbitrary constant)

2026: 18 May Shift 1

Match List-I with List-II

List-I Differential equation	List-II Curve satisfying the differential equation
(A) $\frac{d^2y}{dx^2}+y=0$	(I) $x^2+y^2=C$ : Where C is an arbitrary constant
(B) $\frac{dy}{dx}+y=\frac{x^2+y}{x}$	(II) $3e^{-4y}+4e^{3x}=C$ : Where C is an arbitrary constant
(C) $\frac{dy}{dx}+\frac{x}{y}=0$	(III) $y=\sin x+\cos x+C$ : Where C is an arbitrary constant
(D) $\log(dy/dx)=3x+4y$	(IV) $y=x+Cxe^{-x}$ : Where C is an arbitrary constant

Choose the correct answer from the options given below:

2025: 14 May Shift 1

The solution of the differential equation $log_e\left(\frac{dy}{dx}\right) = 3x + 4y$ is given by

2026: 29 May Shift 2

In reference to the differential equation : $y''+y=(e)^{y'}$ , which of the following statements are correct ?

A. The order of the differential equation is 1.

B. The degree of the differential equation is 1.

C. The order of the differential equation is 2.

D. The degree of the differential equation is not defined.

Choose the correct answer from the options given below:

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]