CBSE Maths › Class 12 PYQs › Differential Equations

Differential Equations: 1 mark Questions (CBSE Class 12)

18 different 1 mark questions on Differential Equations from CBSE Class 12 Maths board exams 2026, newest first.

Q111 markMCQDifferential Equations CBSE 2026 · 65/1/1

The integrating factor of differential equation $R \frac{d x}{d y} + P x = Q$ where P, Q, R are functions of y is

(A) $e^{\int \frac{P}{Q} d y}$
(B) $e^{\int P d y}$
(C) $e^{\int \frac{P}{R} d y}$
(D) $e^{\int \frac{P}{R} d x}$

Show answer & solution

Answer: (C)

e^{\int \frac{P}{R} d y}

Divide by R: $\frac{d x}{d y} + \frac{P}{R} x = \frac{Q}{R}$ , a linear equation in $x$ .
Integrating factor $= e^{\int \frac{P}{R} d y}$ .

Also asked in: 2026 65/1/2, 2026 65/1/3

Q121 markMCQDifferential Equations CBSE 2026 · 65/1/1

The order and degree of the differential equation $\frac{d}{d x} (e^{y}) = 0$ respectively are

(A)0, 1
(B)1, 1
(C)2, 1
(D)1, not defined

Show answer & solution

Answer: (B) 1, 1

$\frac{d}{d x} (e^{y}) = e^{y} \frac{d y}{d x} = 0$ .
The highest derivative is $\frac{d y}{d x}$ , so order $= 1$ .
It appears with power 1 (the equation is a polynomial in the derivative), so degree $= 1$ .

Q51 markMCQDifferential Equations CBSE 2026 · 65/1/2

The order and degree of the differential equation : $\frac{d}{d x} (sin y) = y^{2}$ respectively are

(A)1, 1
(B)2, 1
(C)2, 2
(D)1, 2

Show answer & solution

Answer: (A) 1, 1

$\frac{d}{d x} (sin y) = cos y \frac{d y}{d x} = y^{2}$ .
Highest derivative is $\frac{d y}{d x}$ , so order $= 1$ ; its power is 1, so degree $= 1$ .

Q161 markMCQDifferential Equations CBSE 2026 · 65/1/3

The order and degree of the differential equation :
$\frac{d}{d x} (y^{'})^{3} + (y^{'})^{3} = 1$ respectively are where $y^{'} = \frac{d y}{d x}$

(A)1, 3
(B)2, 1
(C)3, 1
(D)3, 2

Show answer & solution

Answer: (B) 2, 1

$\frac{d}{d x} (y^{'})^{3} = 3 (y^{'})^{2} y^{''}$ , so the equation is $3 (y^{'})^{2} y^{''} + (y^{'})^{3} = 1$ .
Highest derivative is $y^{''}$ , so order $= 2$ .
$y^{''}$ appears with power 1, so degree $= 1$ .

Q111 markMCQDifferential Equations CBSE 2026 · 65/2/1

The general solution of the differential equation $\frac{d y}{d x} = \frac{y}{x}$ is

(A) $lo g y = lo g x + C$
(B) $y + x = C$
(C) $y - x = C$
(D) $lo g y + lo g x = C$

Show answer & solution

Answer: (C)

y - x = C

Separating: $\frac{d y}{y} = \frac{d x}{x}$ .
Integrating: $2 y = 2 x + C_{1}$ .
$y - x = C$ .

Q121 markMCQDifferential Equations CBSE 2026 · 65/2/1

The integrating factor of the differential equation $2 x \frac{d y}{d x} - y = 3$ is

(A) $x$
(B) $\frac{1}{x}$
(C) $e^{x}$
(D) $e^{- x}$

Show answer & solution

Answer: (B)

\frac{1}{x}

Standard form: $\frac{d y}{d x} - \frac{1}{2 x} y = \frac{3}{2 x}$ , so $P = - \frac{1}{2 x}$ .
I.F. $= e^{\int - \frac{1}{2 x} d x} = e^{- \frac{1}{2} l o g x} = \frac{1}{x}$ .

Also asked in: 2026 65/2/2, 2026 65/2/3

Q41 markMCQDifferential Equations CBSE 2026 · 65/2/2

The general solution of the differential equation $x d y - y d x = 0$ is

(A) $x^{2} - y^{2} = k$
(B) $x y = k$
(C) $x = k y$
(D) $lo g y + lo g x = k$

Show answer & solution

Answer: (C)

x = k y

Separating: $\frac{d y}{y} = \frac{d x}{x}$ .
Integrating: $lo g ∣ y ∣ = lo g ∣ x ∣ + lo g ∣ c ∣$ , so $y = c x$ .
Equivalently $x = k y$ (with $k = \frac{1}{c}$ ).

Q151 markMCQDifferential Equations CBSE 2026 · 65/2/3

The general solution of the differential equation : $x^{2} d y + y^{2} d x = 0$ is

(A) $x^{3} + y^{3} = k$
(B) $\frac{1}{y} - \frac{1}{x} = k$
(C) $\frac{1}{y} + \frac{1}{x} = k$
(D) $lo g y^{2} + lo g x^{2} = k$

Show answer & solution

Answer: (C)

\frac{1}{y} + \frac{1}{x} = k

Separating: $\frac{d y}{y ^{2}} = - \frac{d x}{x ^{2}}$ .
Integrating: $- \frac{1}{y} = \frac{1}{x} - k$ .
$\frac{1}{y} + \frac{1}{x} = k$ .

Q131 markMCQDifferential Equations CBSE 2026 · 65/3/1

$\frac{d y}{d x} = F (x, y)$ will be a homogeneous differential equation for which of the following functions ?
(i) $F (x, y) = 3 x + 2 y$
(ii) $F (x, y) = sin \frac{y}{x} + lo g y - lo g x$
(iii) $F (x, y) = e^{y / x} + 1$
(iv) $F (x, y) = x^{2} + y^{2} - y$

(A)(i) and (ii)
(B)(i), (ii) and (iii)
(C)(ii), (iii) and (iv)
(D)(ii) and (iii)

Show answer & solution

Answer: (D) (ii) and (iii)

$\frac{d y}{d x} = F (x, y)$ is homogeneous when F is homogeneous of degree zero, i.e. $F (λ x, λ y) = F (x, y)$ .
(i) $3 x + 2 y$ has degree 1: not homogeneous of degree zero.
(ii) $sin \frac{y}{x} + lo g \frac{y}{x}$ : degree zero.
(iii) $e^{y / x} + 1$ : degree zero.
(iv) $x^{2} + y^{2} - y$ has degree 1.
So (ii) and (iii).

Also asked in: 2026 65/3/2, 2026 65/3/3

Q191 markAssertion–ReasonDifferential Equations CBSE 2026 · 65/3/1

Assertion (A) : One of the particular solutions of the differential equation $\frac{d y}{d x} = e^{x + y}$ can be $e^{x} + e^{- y} = - 2$ .
Reason (R) : $e^{x} + e^{- y} = C$ is the general solution of the differential equation $\frac{d y}{d x} = e^{x + y}$ .

(A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
(B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(C)Assertion (A) is true, but Reason (R) is false.
(D)Assertion (A) is false, but Reason (R) is true.

Show answer & solution

Answer: (D) Assertion (A) is false, but Reason (R) is true.

$e^{- y} d y = e^{x} d x$ gives $- e^{- y} = e^{x} + k$ , i.e. $e^{x} + e^{- y} = C$ . So R is true.
$e^{x} + e^{- y} > 0$ for all real x, y, so it can never equal $- 2$ . A is false.

Also asked in: 2026 65/3/2, 2026 65/3/3

Q121 markMCQDifferential Equations CBSE 2026 · 65/4/1

Product of the order and degree of differential equation $1 + (\frac{d y}{d x})^{3} = λ (\frac{d ^{3} y}{d x ^{3}})^{2}$ is :

(A)5
(B)6
(C)2
(D)3

Show answer & solution

Answer: (B) 6

Highest order derivative is $\frac{d ^{3} y}{d x ^{3}}$ , so order $= 3$ .
It is a polynomial in derivatives and the power of $\frac{d ^{3} y}{d x ^{3}}$ is 2, so degree $= 2$ .
Product $= 3 \times 2 = 6$ .

Q131 markMCQDifferential Equations CBSE 2026 · 65/4/1

Which of the following is not a Linear Differential Equation ?

(A) $(1 + x^{2}) d y + 2 x y d x = cot x d x$
(B) $y + \frac{d}{d x} (x y) = x (sin x + lo g x)$
(C) $x (1 + y^{2}) d x - y (1 + x^{2}) d y = 0$
(D) $y d x - (x + 3 y^{2}) d y = 0$

Show answer & solution

Answer: (C)

x (1 + y^{2}) d x - y (1 + x^{2}) d y = 0

(A): $\frac{d y}{d x} + \frac{2 x}{1 + x ^{2}} y = \frac{c o t x}{1 + x ^{2}}$ , linear in y.
(B): $y + x \frac{d y}{d x} + y = x (sin x + lo g x)$ , linear in y.
(D): $\frac{d x}{d y} - \frac{x}{y} = 3 y$ , linear in x.
(C): $\frac{d y}{d x} = \frac{x ( 1 + y ^{2} )}{y ( 1 + x ^{2} )}$ contains $y^{2}$ , and $\frac{d x}{d y}$ contains $x^{2}$ , so it is not linear.

Also asked in: 2026 65/4/2, 2026 65/4/3

Q121 markMCQDifferential Equations CBSE 2026 · 65/4/2

The sum of the order and the degree of the differential equation $4 [\frac{d ^{2} y}{d x ^{2}}]^{2} + 3 [1 + (\frac{d y}{d x})^{2} - y] = 0$ is :

(A)2
(B)3
(C)not defined
(D)4

Show answer & solution

Answer: (D) 4

Highest order derivative is $\frac{d ^{2} y}{d x ^{2}}$ , so order $= 2$ .
The equation is a polynomial in the derivatives and the highest power of $\frac{d ^{2} y}{d x ^{2}}$ is 2, so degree $= 2$ .
Sum $= 2 + 2 = 4$ .

Q121 markMCQDifferential Equations CBSE 2026 · 65/4/3

The sum of the order and the degree of the differential equation $y = x \frac{d y}{d x} + 3 [2 - (\frac{d y}{d x})^{2}]$ is :

(A)not defined
(B)3
(C)1
(D)2

Show answer & solution

Answer: (B) 3

Only $\frac{d y}{d x}$ appears, so order $= 1$ .
The equation is a polynomial in $\frac{d y}{d x}$ with highest power 2, so degree $= 2$ .
Sum $= 1 + 2 = 3$ .

Q131 markMCQDifferential Equations CBSE 2026 · 65/5/1

The order and degree of the differential equation $1 + (\frac{d ^{3} y}{d x ^{3}})^{3} = λ \frac{d ^{2} y}{d x ^{2}}$ is :

(A)Order = 3, Degree = 3
(B)Order = 2, Degree = 2
(C)Order = 3, Degree = 1
(D)Order = 2, Degree = 1

Show answer & solution

Answer: (A) Order = 3, Degree = 3

Highest derivative is $\frac{d ^{3} y}{d x ^{3}}$ , so order = 3.
It is a polynomial in derivatives and the highest order derivative appears with power 3, so degree = 3.

Q141 markMCQDifferential Equations CBSE 2026 · 65/5/1

The general solution for the differential equation $\frac{d y}{d x} = e^{3 x - y}$ is :

(A) $3 e^{y} = e^{3 x} + C$
(B) $lo g (3 x - y) = C$
(C) $e^{3 x - y} = C$
(D) $- e^{y} + 3 e^{3 x} = C$

Show answer & solution

Answer: (A)

3 e^{y} = e^{3 x} + C

$\frac{d y}{d x} = e^{3 x} e^{- y} \Rightarrow e^{y} d y = e^{3 x} d x$ .
Integrating: $e^{y} = \frac{e ^{3 x}}{3} + C_{1}$ , i.e. $3 e^{y} = e^{3 x} + C$ .

Also asked in: 2026 65/5/2, 2026 65/5/3

Q131 markMCQDifferential Equations CBSE 2026 · 65/5/2

The order and degree of differential equation $\frac{d ^{2} y}{d x ^{2}} = 1 - (\frac{d ^{3} y}{d x ^{3}})^{2}$ is :

(A)Order = 3, Degree = 3
(B)Order = 2, Degree = 2
(C)Order = 3, Degree = 2
(D)Order = 2, Degree = 1

Show answer & solution

Answer: (C) Order = 3, Degree = 2

Highest order derivative is $\frac{d ^{3} y}{d x ^{3}}$ , so order = 3.
The equation is a polynomial in the derivatives and $\frac{d ^{3} y}{d x ^{3}}$ occurs with highest power 2, so degree = 2.

Q131 markMCQDifferential Equations CBSE 2026 · 65/5/3

The order and degree of differential equation $y = (\frac{d ^{2} y}{d x ^{2}})^{2} - λ \frac{d y}{d x}$ is :

(A)Order = 2, Degree = 2
(B)Order = 2, Degree = 3
(C)Order = 1, Degree = 2
(D)Order = 2, Degree = 4

Show answer & solution

Answer: (A) Order = 2, Degree = 2

Highest order derivative is $\frac{d ^{2} y}{d x ^{2}}$ , so order = 2.
The equation is a polynomial in the derivatives and $\frac{d ^{2} y}{d x ^{2}}$ occurs with highest power 2, so degree = 2.

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