Calculus

Differential Equations

Medium

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:

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. $x$

A-III, B-II, C-IV, D-I

A-II, B-I, C-IV, D-III

A-II, B-III, C-I, D-IV

A-II, B-III, C-IV, D-I

✅ Correct Option: 4

A: Particular solution has 0 arbitrary constants (II).

B: General solution of order 2 has 2 constants (III).

C: IF for $\frac{dy}{dx} + \frac{y}{x} = 3$ is $e^{\int dx/x} = x$ (IV).

D: Solving gives $y = 1 + Cx$ , so $\lim_{x \to 0^+} y = 1$ (I).

2026: 18 May Shift 1

Match List-I with List-II

List-I	List-II
Differential equation	Order and degree
(A) $y''' + x^2(y'')^3 + y = 0$	(I) $3, 2$
(B) $y''' + y^2 + e^{y'} = 0$	(II) $3, 3$
(C) $(y''')^2 + x^2(y'')^3 + xy = 0$	(III) $3, 1$
(D) $(y''')^3 + x(y'')^2 + y = 0$	(IV) $3, \text{Not defined}$

Choose the correct answer from the options given below:

2026: 21 May Shift 2

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

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:

2025: 14 May Shift 1

The solution of the differential equation $log_e\left(\frac{dy}{dx}\right) = 3x + 4y$ is given by

2025: 14 May Shift 2

For the differential equation $(x + y)dy + (x - y)dx = 0$ , which of the following is/are correct?

(A) Differential equation is homogeneous

(B) Order of differential equation is 1

(D) Degree of the equation is not defined

Choose the correct answer from the options given below:

2022: 6 Aug Shift 2

If the order and the degree of the differential equation $\left(\frac{dy}{dx}\right)^{\frac{1}{2}} = \left(\frac{d^2y}{dx^2}\right)^{\frac{1}{5}}$ are O and S respectively, then $S - O$ is equal to

Solution

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