Class 12 · Chapter 9 · Calculus unit (35 of 80 marks)

Differential Equations Class 12: notes and important questions

Revision notes, 10 free important questions with step-marked answers, and a four-step route from the basics to 95+ through 50 board-style questions.

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50 questions
14 multiple choice, 4 assertion–reason, 10 very short answer, 13 short answer, 6 long answer, 3 case study
About 16 hours to master

Calculus unit: 35 of 80 theory marks (Continuity and Differentiability, Application of Derivatives, Integrals, Application of Integrals, Differential Equations).

NCERT textbook: Chapter 9 on ncert.nic.in (PDF)

Revision notes

Differential Equations — revision notes

1. Order and degree

Order: order of the highest derivative present.
Degree: power of the highest-order derivative, after the equation is made a polynomial in the derivatives (clear radicals and fractional powers first). If a derivative sits inside $\sin$ , $e^{()}$ , $\ln$ , etc., the degree is not defined.
Order and degree (when defined) are always positive integers.

2. Solutions

General solution: contains as many arbitrary constants as the order. Particular solution: constants fixed by given conditions. To verify a solution, differentiate and substitute.

3. Variables separable: $\frac{d y}{d x} = g (x) h (y)$

$\int \frac{d y}{h (y)} = \int g (x) d x + C$ .

4. Homogeneous: $\frac{d y}{d x} = F (x, y)$ with $F (λ x, λ y) = F (x, y)$

Put $y = v x$ , $\frac{d y}{d x} = v + x \frac{d v}{d x}$ ; the equation becomes separable in $v$ and $x$ . (If $\frac{d x}{d y} = G (x, y)$ is more convenient, put $x = v y$ .) Replace $v$ by $\frac{y}{x}$ at the end.

5. Linear: $\frac{d y}{d x} + P y = Q$ ( $P, Q$ functions of $x$ or constants)

IF $= e^{\int P d x}$ ; solution $y \cdot IF = \int Q \cdot IF d x + C$ . For $\frac{d x}{d y} + P_{1} x = Q_{1}$ : IF $= e^{\int P_{1} d y}$ , $x \cdot IF = \int Q_{1} \cdot IF d y + C$ .

Worked example 1 (separable)

$\frac{d y}{d x} = x y$ : $\ln | y | = \frac{x^{2}}{2} + c \Rightarrow y = C e^{x^{2} / 2}$ .

Worked example 2 (linear)

$x \frac{d y}{d x} + y = 3 x^{2}$ : $\frac{d}{d x} (x y) = 3 x^{2} \Rightarrow x y = x^{3} + C \Rightarrow y = x^{2} + \frac{C}{x}$ .

Worked example 3 (homogeneous)

$\frac{d y}{d x} = \frac{y}{x} + \frac{x}{y}$ : $y = v x \Rightarrow x \frac{d v}{d x} = \frac{1}{v} \Rightarrow \frac{v^{2}}{2} = \ln | x | + C \Rightarrow y^{2} = 2 x^{2} (\ln | x | + C)$ .

Common errors

Stating the degree without first clearing a radical or fractional power.
Forgetting to write the equation in standard form (coefficient of $\frac{d y}{d x}$ equal to $1$ ) before finding $P$ .
Writing $e^{\int P d x}$ with a “ $+ C$ ” inside — the IF needs no constant.
Adding the constant only after dividing: in $y \cdot IF = \int Q \cdot IF d x + C$ , $C$ must be divided by the IF too.
Leaving the answer in $v$ instead of $\frac{y}{x}$ .

Board-exam tips

Say which type the equation is before solving (a step mark often depends on it).
If $\frac{d y}{d x}$ form is not linear in $y$ , try $\frac{d x}{d y}$ — it may be linear in $x$ .
In word problems, write the rate equation with units, and check the answer against the story (e.g. temperature approaches room temperature).

Topics in this chapter: Order and degree · General and particular solutions · Linear differential equations · Homogeneous differential equations · Variables separable · Modelling with differential equations.

Formulas for this chapter: Chapter 9 on the Class 12 formula sheet (free PDF and A3 poster).

Differential Equations Class 12: important questions

10 original questions in the board paper's question types, from 1-mark MCQs to a case study, each with a step-marked worked answer. Try each one first. 5 worked answers open here; the other 5 open with a free account (no card).

Download these questions (PDF) 10 questions, 5 worked answers · 5 pages · 184 KB

Section A style · Multiple choice and assertion–reason (1 mark each)

Q1·1 mark·Multiple choiceOrder and degree

What are the order and the degree of ${(\frac{d^{3} y}{d x^{3}})}^{2} + x {(\frac{d y}{d x})}^{5} = 0$ ?

Show worked answer

The highest derivative is $\frac{d^{3} y}{d x^{3}}$ (order $3$ ), and it is raised to the power $2$ (degree $2$ ) A1
Correct option: (a) order $3$ , degree $2$

Q2·1 mark·Multiple choiceLinear differential equations

The integrating factor of $\frac{d y}{d x} + \frac{2 y}{x} = x^{2}$ ( $x > 0$ ) is

Q3·1 mark·Assertion–reasonVariables separable

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): $y = C e^{3 x}$ is the general solution of $\frac{d y}{d x} = 3 y$ .

Reason (R): $\frac{d y}{d x} = 3 y$ is a first-order linear differential equation.

Section B style · Short answer (2 marks each)

Q4·2 marks·Short answerVariables separable

Solve the differential equation $y \frac{d y}{d x} = x^{2}$ .

Show worked answer

Answer:

3 y^{2} = 2 x^{3} + C

$\begin{aligned} y d y = x^{2} d x \\ \Rightarrow & \frac{y^{2}}{2} = \frac{x^{3}}{3} + c \end{aligned}$ M1
$3 y^{2} = 2 x^{3} + C$ A1

Q5·2 marks·Short answerVariables separable

Find the particular solution of $\frac{d y}{d x} = 2 x e^{- y}$ for which $y = 0$ when $x = 0$ .

Section C style · Short answer (3 marks each)

Q6·3 marks·Short answerLinear differential equations

Solve $\frac{d y}{d x} - y = e^{2 x}$ .

Show worked answer

Answer:

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

Linear with $P = - 1$ : $IF = e^{- x}$ A1
$\begin{aligned} \frac{d}{d x} (y e^{- x}) = e^{2 x} e^{- x} = e^{x} \\ \Rightarrow & y e^{- x} = e^{x} + C \end{aligned}$ M1
$y = e^{2 x} + C e^{x}$ A1

Q7·3 marks·Short answerHomogeneous equations

Solve $\frac{d y}{d x} = \frac{y}{x} + \frac{x}{y}$ ( $x > 0$ ).

Section D style · Long answer (5 marks each)

Q8·5 marks·Long answerLinear differential equations

Find the particular solution of $x \frac{d y}{d x} + y = x^{3}$ given that $y = 1$ when $x = 1$ .

Show worked answer

Answer:

y = \frac{x^{3}}{4} + \frac{3}{4 x}

Standard form: $\frac{d y}{d x} + \frac{y}{x} = x^{2}$ M1
$IF = e^{\int \frac{1}{x} d x} = x$ A1
$\begin{aligned} \frac{d}{d x} (x y) = x^{3} \\ \Rightarrow & x y = \frac{x^{4}}{4} + C \end{aligned}$ M1
$x = 1$ , $y = 1$ : $\begin{aligned} 1 = \frac{1}{4} + C \\ \Rightarrow & C = \frac{3}{4} \end{aligned}$ A1
$y = \frac{x^{3}}{4} + \frac{3}{4 x}$ A1

Q9·5 marks·Long answerHomogeneous equations

Find the particular solution of $x y \frac{d y}{d x} = x^{2} + 2 y^{2}$ given that $y = 0$ when $x = 1$ .

Section E style · Case study (4 marks)

Q10·4 marks·Case studyVariables separable

Bacteria culture. A bacteria culture grows at a rate proportional to its size: $\frac{d P}{d t} = k P$ . It starts with $5000$ bacteria and doubles in $10$ hours.

Solve the differential equation, using $P (0) = 5000$ . [1 mark]
Find $k$ . [1 mark]
After how many hours will there be $40000$ bacteria? [2 marks]

Show worked answer

$\begin{aligned} \frac{d P}{P} = k d t \\ \Rightarrow & P = 5000 e^{k t} \end{aligned}$ A1
$\begin{aligned} 10000 = 5000 e^{10 k} \\ \Rightarrow & k = \frac{\log 2}{10} \end{aligned}$ per hour A1
$\begin{aligned} 5000 e^{k t} = 40000 \\ \Rightarrow & e^{k t} = 8 \\ \Rightarrow & t = \frac{\log 8}{k} = \frac{3 \log 2}{(\log 2) / 10} \end{aligned}$ M1
$t = 30$ hours A1

Next: practise by Route to 95 step · the revision notes · free MCQs and case studies · all Class 12 assertion–reason questions · the Class 12 syllabus checklist.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can find the order and degree of a differential equation and solve simple variables-separable equations.

Read first: 1. Order and degree; 2. Solutions; 3. Variables separable 16 practice questions · checkpoint: 4 questions, 6 marks, pass 80%

Practise step 1

Step 2

Board standard

You can recognise and solve homogeneous and linear equations and find particular solutions from given conditions.

Read first: 3. Variables separable; 4. Homogeneous; 5. Linear; Worked examples 1-3 13 practice questions · checkpoint: 4 questions, 11 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can write complete long answers and case studies, including setting up a differential equation from a real situation.

Read first: 5. Linear; Worked example 2; Board-exam tips 5 practice questions · checkpoint: 3 questions, 14 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can handle equations that are linear in x, mixed-method problems and modelling questions that need interpretation.

Read first: 4. Homogeneous; 5. Linear; Common errors 6 practice questions · checkpoint: 3 questions, 13 marks, pass 80%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 7 of the 50 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceOrder and degree

The sum of the order and the degree of the differential equation ${(\frac{d^{3} y}{d x^{3}})}^{2} + {(\frac{d^{2} y}{d x^{2}})}^{5} + \frac{d y}{d x} = \sin x$ is

(a) $7$
(b) $8$
(c) $5$
(d) $6$

Q2·1 mark·Assertion–reasonOrder and degree

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): The degree of the differential equation ${(\frac{d^{2} y}{d x^{2}})}^{2} + \cos (\frac{d y}{d x}) = 0$ is not defined.

Reason (R): The degree of a differential equation is defined only when the equation is a polynomial in its derivatives.

(a)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(b)Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
(c)Assertion (A) is true but Reason (R) is false.
(d)Assertion (A) is false but Reason (R) is true.

Q3·1 mark·Multiple choiceGeneral and particular solutions

The number of arbitrary constants in the general solution of a differential equation of order $4$ is

(a) $0$
(b) $2$
(c) $3$
(d) $4$

Worked solution·2 marks·Very short answerVariables separable

Find the particular solution of $\frac{d y}{d x} = x (1 + y^{2})$ given that $y = 1$ when $x = 0$ .

Full worked solution Free, no sign-in

$\int \frac{d y}{1 + y^{2}} = \int x d x \Rightarrow \tan^{- 1} y = \frac{x^{2}}{2} + C$ M1
$C = \tan^{- 1} 1 = \frac{π}{4}$ : $y = \tan (\frac{x^{2}}{2} + \frac{π}{4})$ A1

Every question here has a full worked solution. See them all →

Stretch yourself: original calculus and functions problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S132. All 24 →

Knowledge organiser: Differential Equations on one page (key facts, formulas and mistakes to avoid; printable PDF)

Where marks are lost in Differential Equations

Stating the degree before clearing radicals or fractional powers. Fix: make the equation polynomial in the derivatives first; if it cannot be, the degree is not defined.
Finding P before writing the equation as dy/dx + Py = Q. Fix: divide by the coefficient of dy/dx first and write the standard form as its own line.
Dividing only part of the right-hand side by the IF. Fix: write y·IF = ∫Q·IF dx + C, then divide every term (including C) by the IF.
Leaving the homogeneous answer in v. Fix: substitute v = y/x back in the final line.
Not naming the type of equation. Fix: write 'this is homogeneous (or linear, or separable)' before solving; a step mark can depend on it.
Particular solution asked but general solution given. Fix: substitute the given condition to find C and write the final equation.
When dy/dx is not linear in y, giving up. Fix: try dx/dy + Px = Q, which may be linear in x.

Exam strategy: how answers are step-marked · the proofs, written out · competency-based questions

Differential Equations in our sample papers

Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).

Sample paper 1 (Board standard): question 28
Sample paper 2 (Board standard): questions 15, 28, 37
Sample paper 3 (Above board): questions 12, 29
Sample paper 4 (Above board): questions 15, 28
Sample paper 5 (95+ challenge): questions 12, 34

Want it against the clock? Take a timed 30-mark chapter test on Differential Equations, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Preparing for JEE or CUET? Try the JEE and CUET-style questions on integrals, areas and differential equations.

Differential Equations: common questions

What is in CBSE Class 12 Maths Chapter 9, Differential Equations?

The topics are: Order and degree, General and particular solutions, Linear differential equations, Homogeneous differential equations, Variables separable, Modelling with differential equations. This page has revision notes for each, and 50 board-style practice questions with step-marked solutions.

How many marks is Differential Equations worth?

It is in the Calculus unit, which carries 35 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Differential Equations?

Stating the degree before clearing radicals or fractional powers. Fix: make the equation polynomial in the derivatives first; if it cannot be, the degree is not defined. Finding P before writing the equation as dy/dx + Py = Q. Fix: divide by the coefficient of dy/dx first and write the standard form as its own line.

Are the Differential Equations important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

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Revision notes

Differential Equations — revision notes

1. Order and degree

2. Solutions

3. Variables separable: dydx=g(x)h(y)

4. Homogeneous: dydx=F(x,y) with F(λx,λy)=F(x,y)

5. Linear: dydx+Py=Q (P,Q functions of x or constants)

Worked example 1 (separable)

Worked example 2 (linear)

Worked example 3 (homogeneous)

Common errors

Board-exam tips

Differential Equations Class 12: important questions

Section A style · Multiple choice and assertion–reason (1 mark each)

Section B style · Short answer (2 marks each)

Section C style · Short answer (3 marks each)

Section D style · Long answer (5 marks each)

Section E style · Case study (4 marks)

Route to 95: four steps

Secure the basics

Board standard

Full marks on long answers

95+ stretch (HOTS)

Practice questions

Full worked solution Free, no sign-in

Where marks are lost in Differential Equations

Differential Equations in our sample papers

Differential Equations: common questions

Work through the Route to 95 with your progress saved

3. Variables separable: $\frac{d y}{d x} = g (x) h (y)$

4. Homogeneous: $\frac{d y}{d x} = F (x, y)$ with $F (λ x, λ y) = F (x, y)$

5. Linear: $\frac{d y}{d x} + P y = Q$ ( $P, Q$ functions of $x$ or constants)