The solution of the differential equation dy/dx = x^2y^2 with y = 1 at x = 0 is Option 1: y(x^3-3)-3=0 Option 2: y(x^3+3)-3=0 Option 3: y(x^3+3)+3=0 Option 4: y(x^3-3)+3=0

Answer: y(x^3-3)+3=0. Solution: Separating variables, y^(-2)dy= x^2dx gives -1y=x^3/3+C. Using y=1 at x=0, C=-1. So -3y=x^3-3, i.e. y(x^3-3)+3=0.

Calculus

Differential Equations

Medium

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

$y(x^3-3)-3=0$

$y(x^3+3)-3=0$

$y(x^3+3)+3=0$

$y(x^3-3)+3=0$

✅ Correct Option: 4

Separating variables, $\displaystyle\int y^{-2}dy=\int x^2dx$ gives $-\dfrac1y=\dfrac{x^3}{3}+C$ . Using $y=1$ at $x=0$ , $C=-1$ . So $-\dfrac3y=x^3-3$ , i.e. $y(x^3-3)+3=0$ .

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Match the LIST-I with LIST-II

	LIST-I Differential equation		LIST-II Integrating factor
A.	$\dfrac{dy}{dx} + 3y = e^{-2x}$	I.	$\dfrac{1}{x}$
B.	$x\dfrac{dy}{dx} + y = 3x^2$	II.	$e^{-x}$
C.	$x\dfrac{dy}{dx} - y = 3x^2$	III.	$x$
D.	$\dfrac{dy}{dx} - y = x$	IV.	$e^{3x}$

Choose the correct answer from the options given below:

2022: 16 July Shift 2

Integrating factor of the differential equation $\frac{dy}{dx} - \frac{1}{x} y = 1$ is:

The solution of the differential equation dydx=x2y2\dfrac{dy}{dx} = x^2y^2dxdy​=x2y2 with y=1y = 1y=1 at x=0x = 0x=0 is

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