The general solution of the differential equation dy/dx = -4xy^2 is given by Option 1: 2x^2 - y = C; C is an arbitrary constant Option 2: 2x^2 - 1/y = C; C is an arbitrary constant Option 3: 2x^2 + 1/y^2 = C; C is an arbitrary constant Option 4: 2x^2 + 1/y = C; C is an arbitrary constant

Answer: 2x^2 - 1/y = C; C is an arbitrary constant. Solution: The differential equation is dy/dx = -4xy^2 This is a separable differential equation. dy/y^2 = -4x dx Integrating both sides: 1/y^2 dy = -4x dx y^(-2) dy = -4x dx -1/y = -2x^2 + C Rearranging: -1/y = -2x^2 + C 2x^2 - 1/y = C Therefore, the general solution is 2x^2 - 1/y = C

Calculus

Differential Equations

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The general solution of the differential equation $\frac{dy}{dx} = -4xy^2$ is given by

$2x^2 - y = C$ ; C is an arbitrary constant

$2x^2 - \frac{1}{y} = C$ ; C is an arbitrary constant

$2x^2 + \frac{1}{y^2} = C$ ; C is an arbitrary constant

$2x^2 + \frac{1}{y} = C$ ; C is an arbitrary constant

✅ Correct Option: 2

The differential equation is $\frac{dy}{dx} = -4xy^2$

This is a separable differential equation.

$\frac{dy}{y^2} = -4x \, dx$

Integrating both sides:

$\int \frac{1}{y^2} \, dy = \int -4x \, dx$

$\int y^{-2} \, dy = \int -4x \, dx$

$-\frac{1}{y} = -2x^2 + C$

Rearranging:

$-\frac{1}{y} = -2x^2 + C$

$2x^2 - \frac{1}{y} = C$

Therefore, the general solution is $2x^2 - \frac{1}{y} = C$

2026: 21 May Shift 1

For the differential equation $\frac{\left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{3}{2}}}{\frac{d^2y}{dx^2}} = k, k \neq 0$ , which one of the following is TRUE ?

2025: 21 May Shift 2

Match List-I with List-II

List-I	List-II
(A) Degree of the differential equation $\frac{d^2y}{dx^2} = e^{dy/dx}$ is	(I) 2
(B) Order of the differential equation $(\frac{dy}{dx})^2 + \frac{d^3y}{dx^3} = 0$ is	(II) not defined
(C) Degree of the differential equation $\frac{d^2y}{dx^2} + (\frac{dy}{dx})^2 - 5x^2 = 0$	(III) 3
(D) If p is the order and q is the degree of the differential equation $\frac{dy}{dx} + 3y = e^x$ , then p + q is	(IV) 1

Choose the correct answer from the options given below:

2023: 30 May Shift 3

The degree of the differential equation $\left[1 + \left(\frac{dy}{dx}\right)\right]^3 = \left(\frac{d^2 y}{dx^2}\right)^2$ is :

2023: 22 May Shift 3

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

2025: 30 May Shift 1

The differential equation representing the family of curves $y = Ax + \frac{B}{x}$ , $x \neq 0$ where A and B are arbitrary constants, is given by

2025: 27 May Shift 1

Match List-I with List-II

List-I	List-II
(A) The degree of the differential equation $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = x\sin\left(\frac{dy}{dx}\right)$	(I) 4
(B) The degree of differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^{1/4} + x^{1/5} = 0$	(II) 1
(C) The degree of differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 + 6y^5 = 0$	(III) Not defined
(D) The degree of differential equation $1 + \left(\frac{dy}{dx}\right)^4 = 7\left(\frac{d^2y}{dx^2}\right)^3$	(IV) 3

Choose the correct answer from the options given below:

The general solution of the differential equation dydx=−4xy2\frac{dy}{dx} = -4xy^2dxdy​=−4xy2 is given by

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The general solution of the differential equation $\frac{dy}{dx} = -4xy^2$ is given by