Differential Equations MCQ with Solutions

Differential Equations: Chapter 9 Links

Current page: Differential Equations MCQ. Go to other exercises and content from this chapter :

9.1 | 9.2 | 9.3 | 9.4 | 9.5 | Misc.

Differential Equations MCQ with answers and solutions from NCERT Class 12 Maths. Revise all 12 important questions step by step with clear explanations. You can also watch the video solutions for better understanding. This is in continuation of the previous topic on Application of Integrals MCQs.

Ready? Okay, let’s begin with the first question!

Differential Equations MCQ#1 (NCERT Exercise 9.1 – Question 12)

📍The order of the differential equation $2x^{2}\dfrac{d^{2}y}{dx^{2}}-3\dfrac{dy}{dx}+y=0$ is

Options:

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

Answer:
✅ Correct option: (A)

Explanation:
Order of a differential equation is the highest order derivative present in it.

Here, the highest derivative is $\dfrac{d^{2}y}{dx^{2}}$ .

Hence, the order of the differential equation is $2$ . Do watch the video below for the concept along with some more such questions.

Watch Video Solution on YouTube

Differential Equations MCQ#2 (NCERT Exercise 9.1 – Question 11)

📍The degree of the differential equation $\left(\dfrac{d^{2}y}{dx^{2}}\right)^{3}+\left(\dfrac{dy}{dx}\right)^{2}+\sin\left(\dfrac{dy}{dx}\right)+1=0$ is

Options:

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

Answer:
✅ Correct option: (D)

Explanation:
Degree is defined only when the differential equation is a polynomial in its derivatives.

Here, $\sin\left(\dfrac{dy}{dx}\right)$ is present, which is a trigonometric function of derivative.

Hence, the equation is not polynomial in derivatives.

Therefore, degree is not defined. Do watch the video below for the concept along with a few more similar questions.

Watch Video Solution on YouTube

Let me tell you all of these 12 Differential Equations MCQ are very simple and easy to understand and therefore scoring too. But you must practice them regularly.

So, my suggestion to you is to solve these NCERT MCQs a good number of times and feel confident. And if you ever need any clarification, just drop a comment — I’ll be more than happy to help.

Differential Equations MCQ#3 (NCERT Exercise 9.2 – Question 11)

📍The number of arbitrary constants in the general solution of a differential equation of fourth order are:

Options:

(A) $0$
(B) $2$
(C) $3$
(D) $4$

Answer:
✅ Correct option: (D)

Explanation:
The number of arbitrary constants in the general solution of a differential equation is equal to its order.

Since the order is $4$ , the general solution contains $4$ arbitrary constants.

Watch Video Solution on YouTube

Differential Equations MCQ#4 (NCERT Exercise 9.2 – Question 12)

📍The number of arbitrary constants in the particular solution of a differential equation of third order are:

Options:

(A) $3$
(B) $2$
(C) $1$
(D) $0$

Answer:
✅ Correct option: (D)

Explanation:
A particular solution is obtained after assigning specific values to all arbitrary constants.

Hence, a particular solution contains $0$ arbitrary constants.

Watch Video Solution on YouTube

Following is the next Differential Equations MCQ in this part of the series.

Differential Equation MCQ#5 (NCERT Exercise 9.3 – Question 23)

📍The general solution of the differential equation $\dfrac{dy}{dx}=e^{x+y}$ is

Options:

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

Answer:
✅ Correct option: (A)

Explanation:
Given $\dfrac{dy}{dx}=e^{x+y}$

$=e^{x}e^{y}$

Separate the variables:
$e^{-y}dy=e^{x}dx$

Integrate both sides:
$\int e^{-y}dy=\int e^{x}dx$

$-e^{-y}=e^{x}+C$

Rearranging,
$e^{x}+e^{-y}=C$

Hence, correct option is (A).

Video Solution on YouTube

Did you know, there are about 100 MCQs in NCERT textbooks for Class 12 (both Part 1 and Part 2)? I shall be taking one Chapter at a time and show you the explanations for each of the questions.

Moreover, I have also covered these solutions in the video form and they are freely available on my YouTube channel, @Mathsbetter. And here also, in each of the 12 Differential Equations MCQ on this page, I have provided the direct links to each and every MCQ for you to understand the solutions in a very simple and easy manner.

Differential Equations MCQ#6 (NCERT Exercise 9.4 – Question 16)

📍A homogeneous differential equation of the form $\dfrac{dx}{dy}=h\left(\dfrac{x}{y}\right)$ can be solved by making the substitution

Options:

(A) $y=vx$
(B) $v=yx$
(C) $x=vy$
(D) $x=v$

Answer:
✅ Correct option: (C)

Explanation:
Given equation is of the form $\dfrac{dx}{dy}=h\left(\dfrac{x}{y}\right)$ .

Such equations are solved by putting
$\dfrac{x}{y}=v \Rightarrow x=vy$ .

Hence, correct substitution is $x=vy$ .

Watch Video Solution on Yoube

Differential Equation MCQ#7 (NCERT Exercise 9.4 – Question 17)

📍Which of the following is a homogeneous differential equation?

Options:

(A) $(4x+6y+5)\,dy-(3y+2x+4)\,dx=0$
(B) $(xy)\,dx-(x^{3}+y^{3})\,dy=0$
(C) $(x^{3}+2y^{2})\,dx+2xy\,dy=0$
(D) $y^{2}\,dx+(x^{2}-xy-y^{2})\,dy=0$

Answer:
✅ Correct option: (D)

Explanation:
A differential equation $M(x,y)dx+N(x,y)dy=0$ is homogeneous if $M$ and $N$ are homogeneous functions of the same degree.

In option (D):
$M=y^{2}$ (degree 2)
$N=x^{2}-xy-y^{2}$ (degree 2)

Both are of same degree.

Hence, option (D) is homogeneous.

Watch Video Solution on YouTube

Differential Equations MCQ#8 (NCERT Exercise 9.5 – Question 18)

📍The Integrating Factor of the differential equation $x\dfrac{dy}{dx}-y=2x^{2}$ is

Options:

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

Answer:
✅ Correct option: (C)

Explanation:
Given equation:
$x\dfrac{dy}{dx}-y=2x^{2}$

Divide throughout by $x$ :
$\dfrac{dy}{dx}-\dfrac{y}{x}=2x$

This is linear with
$P(x)=-\dfrac{1}{x}$

Integrating Factor (I.F.) is
$e^{\int P(x)\,dx}=e^{\int -\frac{1}{x}dx}=e^{-\ln x}=\dfrac{1}{x}$

Hence, correct option is (C).

Watch Video Solution on YouTube

Differential Equations MCQ#9 (NCERT Exercise 9.5 – Question 19)

📍The Integrating Factor of the differential equation $(1-y^{2})\dfrac{dx}{dy}+yx=ay \;(-1<y<1)$ is

Options:

(A) $\dfrac{1}{y^{2}-1}$
(B) $\dfrac{1}{\sqrt{y^{2}-1}}$
(C) $\dfrac{1}{1-y^{2}}$
(D) $\dfrac{1}{\sqrt{1-y^{2}}}$

Answer:
✅ Correct option: (D)

Explanation:
Given:
$(1-y^{2})\dfrac{dx}{dy}+yx=ay$

Divide by $(1-y^{2})$ :
$\dfrac{dx}{dy}+\dfrac{y}{1-y^{2}}x=\dfrac{ay}{1-y^{2}}$

Here,
$P(y)=\dfrac{y}{1-y^{2}}$

Integrating Factor is
$e^{\int P(y)\,dy}=e^{\int \frac{y}{1-y^{2}}dy}$

Let $t=1-y^{2} \Rightarrow dt=-2y\,dy$

$\int \frac{y}{1-y^{2}}dy=-\dfrac{1}{2}\ln(1-y^{2})$

Thus I.F.
$=e^{-\frac{1}{2}\ln(1-y^{2})}=\dfrac{1}{\sqrt{1-y^{2}}}$

Hence, correct option is (D).

Watch Video Solution on YouTube

You may have been following this website as a resource for some of the important questions. For example, how to Integrate Square Root of tan x, or may be a complete guide to look at using Matrix Method in solving a system of linear equations.

Then trust me, similarly, this series on NCERT Class 12 MCQs is going to help you in many ways.

Differential Equation MCQ#10 (NCERT Misc. Exercise Chapter 9 – Question 13)

📍The general solution of the differential equation $\dfrac{y\,dx-x\,dy}{y}=0$ is

Options:

(A) $xy=C$
(B) $x=Cy^{2}$
(C) $y=Cx$
(D) $y=Cx^{2}$

Answer:
✅ Correct option: (C)

Explanation:
Given
$\dfrac{y\,dx-x\,dy}{y}=0$

$\Rightarrow y\,dx-x\,dy=0$

Divide by $y^{2}$ :
$\dfrac{y\,dx-x\,dy}{y^{2}}=0$

$\Rightarrow d\left(\dfrac{x}{y}\right)=0$

Hence,
$\dfrac{x}{y}=C \Rightarrow y=Cx$ .

Watch Video Solution on YouTube

Differential Equations MCQ#11 (NCERT Misc. Exercise Chapter 9 – Question 14)

📍The general solution of a differential equation of the type $\dfrac{dx}{dy}+P_{1}x=Q_{1}$ is

Options:

(A) $y e^{\int P_{1}dy}=\int (Q_{1}e^{\int P_{1}dy})\,dy+C$
(B) $y e^{\int P_{1}dx}=\int (Q_{1}e^{\int P_{1}dx})\,dx+C$
(C) $x e^{\int P_{1}dy}=\int (Q_{1}e^{\int P_{1}dy})\,dy+C$
(D) $x e^{\int P_{1}dx}=\int (Q_{1}e^{\int P_{1}dx})\,dx+C$

Answer:
✅ Correct option: (C)

Explanation:
The given equation is linear in $x$ with independent variable $y$ .

Integrating Factor is
$e^{\int P_{1}dy}$

Hence the solution is
$x e^{\int P_{1}dy}=\int (Q_{1}e^{\int P_{1}dy})\,dy+C$ .

Watch Video Solution on YouTube

Differential Equations MCQ#12 (NCERT Misc. Exercise Chapter 9 – Question 15)

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

Options:

(A) $x e^{y}+x^{2}=C$
(B) $x e^{y}+y^{2}=C$
(C) $y e^{x}+x^{2}=C$
(D) $y e^{y}+x^{2}=C$

Answer:
✅ Correct option: (C)

Explanation:
Given equation:
$e^{x}dy+(y e^{x}+2x)\,dx=0$

First divide throughout by $e^{x}$ to simplify:

$dy+\left(y+\frac{2x}{e^{x}}\right)dx=0$

$dy+y\,dx=-\frac{2x}{e^{x}}dx$

Now write in standard linear form:
$\dfrac{dy}{dx}+y=-2x e^{-x}$

Here,
$P(x)=1$

Integrating Factor (I.F.) is:
$I.F.=e^{\int 1\,dx}=e^{x}$

Multiply both sides by $e^{x}$ :

$e^{x}\dfrac{dy}{dx}+y e^{x}=-2x$

Left side becomes derivative of product:
$\dfrac{d}{dx}(y e^{x})=-2x$

Integrate both sides:
$\int \dfrac{d}{dx}(y e^{x})\,dx=\int -2x\,dx$

$y e^{x}=-x^{2}+C$

Rearranging constant:
$y e^{x}+x^{2}=C$

Hence, general solution is $y e^{x}+x^{2}=C$ .

Watch Video Solution on YouTube

Quick Answer Key

Q1 → (A)
Q2 → (D)
Q3 → (D)
Q4 → (D)
Q5 → (A)
Q6 → (C)
Q7 → (D)
Q8 → (C)
Q9 → (D)
Q10 → (C)
Q11 → (C)
Q12 → (C)

Closing Note

That completes all the important NCERT Class 12 Maths Differential Equations MCQ from chapter 9.
I hope the explanations and video solutions made things clearer and gave you more confidence. Stay tuned for the next chapter, where we’ll cover all the important Vector Algebra MCQ (Chapter 10 from NCERT) with the same clarity and video support.

👉 Make sure to practice these questions again to strengthen your concepts.
👉 Watch the video solutions for a clearer and faster revision.

Keep practicing, and you’ll master the differential equations easily! 🚀

Differential Equations MCQ with Solutions

Differential Equations: Chapter 9 Links

Differential Equations MCQ#1 (NCERT Exercise 9.1 – Question 12)

Differential Equations MCQ#2 (NCERT Exercise 9.1 – Question 11)

Differential Equations MCQ#3 (NCERT Exercise 9.2 – Question 11)

Differential Equations MCQ#4 (NCERT Exercise 9.2 – Question 12)

Differential Equation MCQ#5 (NCERT Exercise 9.3 – Question 23)

Differential Equations MCQ#6 (NCERT Exercise 9.4 – Question 16)

Differential Equation MCQ#7 (NCERT Exercise 9.4 – Question 17)

Differential Equations MCQ#8 (NCERT Exercise 9.5 – Question 18)

Differential Equations MCQ#9 (NCERT Exercise 9.5 – Question 19)

Differential Equation MCQ#10 (NCERT Misc. Exercise Chapter 9 – Question 13)

Differential Equations MCQ#11 (NCERT Misc. Exercise Chapter 9 – Question 14)

Differential Equations MCQ#12 (NCERT Misc. Exercise Chapter 9 – Question 15)

Quick Answer Key

Closing Note

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Differential Equations: Chapter 9 Links

Differential Equations MCQ#1 (NCERT Exercise 9.1 – Question 12)

Differential Equations MCQ#2 (NCERT Exercise 9.1 – Question 11)

Differential Equations MCQ#3 (NCERT Exercise 9.2 – Question 11)

Differential Equations MCQ#4 (NCERT Exercise 9.2 – Question 12)

Differential Equation MCQ#5 (NCERT Exercise 9.3 – Question 23)

Differential Equations MCQ#6 (NCERT Exercise 9.4 – Question 16)

Differential Equation MCQ#7 (NCERT Exercise 9.4 – Question 17)

Differential Equations MCQ#8 (NCERT Exercise 9.5 – Question 18)

Differential Equations MCQ#9 (NCERT Exercise 9.5 – Question 19)

Differential Equation MCQ#10 (NCERT Misc. Exercise Chapter 9 – Question 13)

Differential Equations MCQ#11 (NCERT Misc. Exercise Chapter 9 – Question 14)

Differential Equations MCQ#12 (NCERT Misc. Exercise Chapter 9 – Question 15)

Quick Answer Key

Closing Note

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