Calculus

Differential Equations

Medium

applied

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:

A and B only

C and D only

B and C only

A and D only

✅ Correct Option: 2

The highest derivative is $y''$ , so the order is 2 (C true). Since $y'$ appears inside an exponential, the equation is not a polynomial in its derivatives, so the degree is not defined (D true).

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:

2022: 7 Aug Shift 2

If $U(x) = x + \sqrt{1 + x^2}$ , then solution of the differential equation $\frac{dy}{dx} + \sqrt{\frac{1 + y^2}{1 + x^2}} = 0$ , is :

2026: 12 May Shift 2

Match List-I with List-II

List-I (Differential Equation)	List-II (Order and degree)
(A) $\left(\dfrac{d^2y}{dx^2}\right)^2 + e^{\frac{dy}{dx}} = 0$	(I) order = 2, degree = 1
(B) $xy\dfrac{d^2y}{dx^2} + x\left(\dfrac{dy}{dx}\right)^2 - y\dfrac{dy}{dx} = 0$	(II) order = 2, degree = not defined
(C) $\left(\dfrac{d^3y}{dx^3}\right)^2 + \log_e\left(\dfrac{dy}{dx}\right) = 4x$	(III) order = 3, degree = 2
(D) $\dfrac{d}{dx}\left(\dfrac{d^2y}{dx^2}\right) + y\dfrac{dy}{dx} = \left(1+\dfrac{dy}{dx}\right)^{1/2}$	(IV) order = 3, degree = not defined

Choose the correct answer from the options given below:

2025: 22 May Shift 1

Match List-I with List-II

List-I	List-II
Differential Equation	Degree
(A) $xy\frac{d^2y}{dx^2} + x\left(\frac{dy}{dx}\right)^2 - y\frac{dy}{dx} = 0$	(I) 3
(B) $\frac{d^2y}{dx^2} + \log\left(\frac{dy}{dx}\right) = 0$	(II) 1
(C) $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 + \frac{dy}{dx} + 1 = 0$	(III) not defined
(D) $2x^2\left(\frac{d^2y}{dx^2}\right)^3 - 5\left(\frac{dy}{dx}\right)^3 + y = 0$	(IV) 2

Choose the correct answer from the options given below:

2026: 12 May Shift 2

Which of the following differential equations has $y=x$ as one of the particular solution?

2025: 21 May Shift 2

The solution of the differential equation $(x + 1)\frac{dy}{dx} + 1 - 2e^{-y} = 0$ , $y(0) = 0$ is

Solution

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