Calculus

Differential Equations

Hard

common

Which of the following differential equation represents the family of circles touching the x-axis at the origin ?

$(x^2 - y^2)dy - 2xy\, dx = 0$

$(x^2 + y^2)dy + 2xy\, dx = 0$

$(x^2 - y^2)dx + 2xy\, dy = 0$

$(x^2 + y^2)dy - 2xy\, dx = 0$

✅ Correct Option: 1

Circles touching x-axis at origin have center $(0, a)$ and radius $|a|$ , so $x^2 + (y-a)^2 = a^2$ , i.e. $x^2 + y^2 - 2ay = 0$ . Thus $a = (x^2+y^2)/(2y)$ . Differentiating $x^2 + y^2 - 2ay = 0$ : $2x + 2yy' = 2ay'$ , so $y' = 2xy/(x^2 - y^2)$ . Hence $(x^2 - y^2)dy - 2xy\,dx = 0$ .

2026: 18 May Shift 1

Solution of differential equation $\frac{dy}{dx} = e^{x+y} - xe^y$ is

2026: 31 May Shift 2

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:

2026: 14 May Shift 2

The solution of the differential equation $\frac{dy}{dx} = x^2 + x + \frac{1}{x}$ is

2026: 26 May Shift 2

If $\dfrac{dy}{dx} = e^{x-y} + x^2e^{-y}$ , then the solution of the differential equation will be:-

2026: 18 May Shift 2

Match the LIST-I with LIST-II

LIST-I Differential equation	LIST-II Order and degree
A. $y'' = [1+(y')^2]^{3/2}$	I. 3, 4
B. $y''' + y' + y = 2x$	II. 2, 2
C. $2(y'')^3 - (y')^4 + y = 0$	III. 3, 1
D. $(y''')^4 + (y'')^3 + y' + y = 0$	IV. 2, 3

Choose the correct answer from the options given below:

2025: 27 May Shift 1

Which one of the following equations is a homogeneous differential equation?

Which of the following differential equation represents the family of circles touching the x-axis at the origin ?

Solution

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