Calculus

Differential Equations

Medium

common

The differential equation whose order and degree are 3 and 2 respectively is

$\left(\frac{d^2y}{dx^2}\right)^2 + e^{\frac{dy}{dx}} = y$

$\left(\frac{d^3y}{dx^3}\right)^2 + \log\left(\frac{dy}{dx}\right) = 4x$

$\frac{d}{dx}\left(\frac{d^2y}{dx^2}\right) + y\frac{dy}{dx} = \left(1+\frac{dy}{dx}\right)^{1/2}$

$\frac{d^3y}{dx^3} + x\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 = \log x$

✅ Correct Option: 3

Degree is defined only when the equation is polynomial in the derivatives, which rules out the options containing $e^{y'}$ and $\log y'$ .

Option 3 reads $y'''+yy'=(1+y')^{1/2}$ ; squaring gives $(y'''+yy')^2=1+y'$ . The highest derivative is $y'''$ and its power is $2$ , so order $3$ and degree $2$ .

2022: 6 Aug Shift 2

Match List I with List II

List I	List II
A. The number of arbitrary constants in the particular solution of differential equation of order 2	I. 1
B. The number of arbitrary constants in the general solution of differential equation of order 2	II. 0
C. The integrating factor of differential equation $\frac{dy}{dx} + \frac{1}{x}y = 3, x > 0$ , is	III. 2
D. For differential equation, $x^2 \frac{dy}{dx} + x = xy, x > 0, \lim_{x \to 0^+} y(x)$ is equal to	IV. $x$

Choose the correct answer from the options given below:

2025: 15 May Shift 2

The general solution of the differential equation $\frac{xdy}{dx} + 4y = x^3, (x \neq 0)$ is:

2025: 26 May Shift 2

In the following differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 = 2x^2 \log\left(\frac{d^2y}{dx^2}\right)$ order and degree is:

2025: 21 May Shift 1

Match List-I with List-II

List-I	List-II
Differential Equation	General solution
---	---
(A) $\dfrac{dy}{dx} = \dfrac{y}{x}; x \neq 0$	(I) $y = cx; c \text{ is an arbitrary constant}$
(B) $x dx - y dy = 0; y \neq 0, x \neq 0$	(II) $x^2 - y^2 = c; c \text{ is an arbitrary constant}$
(C) $\dfrac{(x^2 - 1)}{y^2 + 1}\dfrac{dx}{dy} = 1$	(III) $2x + 3y = c; c \text{ is an arbitrary constant}$
(D) $2 dx + 3 dy = 0$	(IV) $(x^3-y^3) = c + 3(x+y); c \text{ is an arbitrary constant}$

Choose the correct answer from the options given below:

2023: 22 May Shift 3

Integerating factor of $(x \log_e x) \frac{dy}{dx} + y = 2 \log_e x$ is :

2026: 25 May Shift 1

General solution of the differential equation $(e^x + e^{-x})dy = (3e^{2x} + 3e^{4x})dx$ is:

[Where C is an arbitrary constant]

The differential equation whose order and degree are 3 and 2 respectively is

Solution

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