Calculus

Differential Equations

Easy

applied

The order and degree of the differential equation $\dfrac{d^3y}{dx^3} + \left(\dfrac{d^2y}{dx^2}\right)^2 + e^{\frac{dy}{dx}} = 2$ is

Order = 2 ; degree = 2

Order = 3 ; degree = 1

Order = 3 ; degree is not defined

Order = 1 ; degree = 3

✅ Correct Option: 3

The highest derivative present is $\dfrac{d^3y}{dx^3}$ , so the order is $3$ .

Degree is defined only when the equation is a polynomial in all derivatives.

The term $e^{dy/dx}$ is not polynomial in $\dfrac{dy}{dx}$ , so the degree is not defined.

2025: 15 May Shift 1

Match List-I with List-II

List-I	List-II
(Differential equation)	(Order and Degree)
(A) $\frac{d^3y}{dx^3} + y^2 + e^{dy/dx} = 0$	(I) order = 3, degree = 1
(B) $\left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} + 1 = 0$	(II) order = 3, degree not defined
(C) $2x^2\frac{d^2y}{dx^2} - 3\left(\frac{dy}{dx}\right)^2 + y = 0$	(III) order = 2, degree = 3
(D) $\frac{d^3y}{dx^3} + 2\left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} = 0$	(IV) order = 2, degree = 1

Choose the correct answer from the options given below:

2025: 3 June Shift 1

The sum of order and degree of the differential equation $(x^2\frac{d^2y}{dx^2})^{3/4} = 5(\frac{dy}{dx})^2 - 3$ is equal to

2025: 21 May Shift 2

Match List-I with List-II

List-I	List-II
(A) Integrating factor of $xdy - (y + x^2)dx = 0$	(I) $x^2$
(B) Integrating factor of $xdy + (2y + x^2)dx = 0$	(II) $x^3$
(C) Integrating factor of $(3y - x^2)dx + xdy = 0$	(III) $x$
(D) Integrating factor of $(y + 3x^2)dx + xdy = 0$	(IV) $\frac{1}{x}$

Choose the correct answer from the options given below:

2026: 15 May Shift 1

If $c$ is an arbitrary constant, then

Match the LIST-I with LIST-II

	LIST-I (Differential Equation)		LIST-II (Solution)
A.	$\dfrac{dy}{dx} = 1 + x + y + xy$	I.	$\log_e \vert y \vert = 2x + \log_e (x-1)^2 + c$
B.	$(x-1)\dfrac{dy}{dx} = 2xy$	II.	$y = cx$
C.	$\dfrac{dy}{dx} = 1 - x + y - xy$	III.	$\log_e \vert 1+y \vert = x + \dfrac{x^2}{2} + c$
D.	$x\,dy = y\,dx$	IV.	$\log_e \vert 1+y \vert = x - \dfrac{x^2}{2} + c$

Choose the correct answer from the options given below:

2022: 23 Aug Shift 1

Match List - I with List - II

List - I (Differential Equation)	List - II (Degree)
A. $\left[1 + (y')^2\right]^2 = y''$	I. $2$
B. $\left[1 + (y'')^3\right]^{\frac{1}{2}} = (y')^3$	II. $4$
C. $(y''')^2 + y'' + 3y' + 5y = e^x$	III. $1$
D. $\left[1 + (y')^3\right]^{\frac{1}{2}} = (y'')^2$	IV. $3$

Choose the correct answer from the option given below:

2024: 16 May Shift 1

Choose the correct answer from the options given below:

List-I	List-II
(A) Integrating factor of $x \, dy - (y + 2x^2) \, dx = 0$	(I) $\frac{1}{x}$
(B) Integrating factor of $(2x^2 - 3y) \, dx = x \, dy$	(II) $x$
(C) Integrating factor of $(2y + 3x^2) \, dx + x \, dy = 0$	(III) $x^2$
(D) Integrating factor of $2x \, dy + (3x^3 + 2y) \, dx = 0$	(IV) $x^3$

The order and degree of the differential equation d3ydx3+(d2ydx2)2+edydx=2\dfrac{d^3y}{dx^3} + \left(\dfrac{d^2y}{dx^2}\right)^2 + e^{\frac{dy}{dx}} = 2dx3d3y​+(dx2d2y​)2+edxdy​=2 is