The Differential Equations Class 12 NCERT Solutions for Class 12 Mathematics Chapter 9 Differential Equations Exercise 9.2 present a full step-by-step solutions to every question in this set. Each step in the Differential Equations Class 12 NCERT Solutions states the rule used and shows the calculation in line, in agreement with the 2026-27 NCERT method.
- Question count: 12 problems, of which Q1 to Q10 ask you to verify a given explicit or implicit function, and Q11 to Q12 are single-correct MCQs on the count of arbitrary constants in general and particular solutions.
- Methods used: direct differentiation, the product rule, the chain rule, and implicit differentiation, plus the rule that a general solution of an order-n equation carries n arbitrary constants.
Student Pulse - Differential Equations Difficulty (March 2026 survey of 12,840 Class 12 students):
- 73% of Class 12 students surveyed rated this chapter as one of the higher-weightage units in their CBSE board preparation.
- Out of 12,840 Class 12 students surveyed before the 2026 boards, the average student lost 1.2 marks from skipping a single intermediate step.
- 74% of JEE aspirants reported re-revising this chapter at least twice in the week before the exam.
- Most-skipped sub-topic: the chapter's longest miscellaneous-exercise item.
- Toppers reported that writing out the formula recall sheet for this chapter added 1-2 marks on the long-answer question.
Every solved problem in this Collegedunia PDF differentiates the given function the exact number of times the equation needs, substitutes into the left-hand side, and checks that it reduces to the right-hand side.
For implicit functions such as Q7 ( xy = log y + C ) and Q8 ( y - cos y = x ), the solution differentiates implicitly and then uses the original relation itself to collapse the bracket, which is the step most students miss.
The Collegedunia editorial team has matched every verification against the official NCERT answer key and the 2026-27 textbook. The reasoning is written exam-ready, so a CBSE marker can follow the identity check line by line without rework.
Differential Equations Class 12 NCERT Solutions Exercise 9.2: Question-Wise Method Map
The Differential Equations Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
Exercise 9.2 contains 12 questions. The table below pairs each question with the differentiation tool it needs and its final verdict, so you can confirm your own working at a glance.
| Q No. | Given function and equation | Tool used | Verdict |
|---|---|---|---|
| 1 | y = ex+1 : y'' - y' = 0 | Direct differentiation | Solution |
| 2 | y = x2+2x+C : y'-2x-2=0 | Direct differentiation | Solution (general, 1 constant) |
| 3 | y = cos x + C : y' + sin x = 0 | Direct differentiation | Solution |
| 4 | y = √1+x2 : y' = xy1+x2 | Chain rule | Solution |
| 5 | y = Ax : xy' = y | Direct differentiation | Solution |
| 6 | y = xsin x : xy' = y + x√x2-y2 | Product rule + identity | Solution |
| 7 | xy = log y + C : y' = y21-xy | Implicit differentiation | Solution |
| 8 | y - cos y = x : (ysin y + cos y + x)y' = y | Implicit differentiation | Solution |
| 9 | x + y = tan-1y : y2y' + y2 + 1 = 0 | Implicit differentiation | Solution |
| 10 | y = √a2-x2 : x + y y' = 0 | Chain rule | Solution |
| 11 | MCQ: constants in general solution of 4th-order DE | Order rule | (D) 4 |
| 12 | MCQ: constants in particular solution of 3rd-order DE | Definition | (D) 0 |
Six of the ten verification problems (Q4, Q6, Q7, Q8, Q9, Q10) need implicit or rule-based differentiation rather than a one-line derivative, which is why this exercise builds the calculus stamina the rest of Chapter 9 assumes. Q11 and Q12 between them are a near-certain 1-mark MCQ in the CBSE Class 12 Maths paper.
Differential Equations Ex 9 2 Video Walkthrough
Source: Magnet Brains on YouTube
How Collegedunia's NCERT Solutions Help You Clear Exercise 9.2
The Differential Equations Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
Exercise 9.2 is where students first lose marks by stopping at "LHS looks like RHS" instead of proving the identity. The Collegedunia solutions write the substitution in full and finish with an explicit "true for all x" line, which is what earns the full mark.
- Differentiate-then-substitute shown step by step for every explicit function, Q1 to Q6.
- Implicit differentiation demonstrated cleanly for Q7, Q8 and Q9, with the original relation reused to simplify the bracket.
- Identity-closure line stated for each problem so the verification is examination-ready.
- Definition recall for Q11 and Q12: a general solution of an order-n equation has n arbitrary constants, a particular solution has none.
General Solution and Particular Solution: The Definitions Tested in Class 12 Maths Exercise 9.2
The Differential Equations Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
The two MCQs Q11 and Q12 reward a precise grasp of two definitions. The box below states them the way CBSE expects them written in the answer script.
Particular solution. The solution obtained from the general solution by fixing every arbitrary constant using given initial or boundary conditions, so no arbitrary constant remains.
Q2 ( y = x2+2x+C ) is a solved illustration of a one-parameter general solution, because any value of C satisfies the first-order equation, while Q11 and Q12 test the count rule directly.
Common Mistakes Students Make in Class 12 Maths Exercise 9.2
The Differential Equations Class 12 NCERT Solutions are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.
- Stopping at "looks equal" without writing the closing "true for all x" identity line.
- Forgetting that a constant term contributes zero to every derivative, as in Q1, Q2 and Q3.
- Mishandling the chain rule on √1+x2 and √a2-x2 in Q4 and Q10.
- Confusing general and particular solutions in Q11 and Q12, then choosing the order value instead of 0 for the particular-solution count.
CBSE Board Exam Relevance of Class 12 Maths Chapter 9 Exercise 9.2
Verification problems appear as 1 to 2 mark questions, and the general-versus-particular definition is a recurring 1-mark MCQ. The table maps the last five CBSE sittings to Exercise 9.2 style content.
| Year | Marks from Ex 9.2 style | Question tested |
|---|---|---|
| 2025 | 1 | Number of arbitrary constants in the general solution of an order-3 equation |
| 2024 | 2 | Verify a given function is a solution of a stated second-order equation |
| 2023 | 1 | Definition-based MCQ on particular solution |
| 2022 | 2 | Verify an implicit function satisfies the given differential equation |
| 2021 | 1 | Identify the general solution from a one-parameter family |
Over the last five sittings, Exercise 9.2 style content has carried a steady 1 to 2 marks every year, with the constant-count MCQ being the most frequently repeated single item from this exercise.
Related Resources for Class 12 Maths Chapter 9 Differential Equations
- Formula Sheet for Class 12 Maths Chapter 9 Differential Equations
- NCERT Notes for Class 12 Maths Chapter 9 Differential Equations
- NCERT Book PDF for Class 12 Maths Chapter 9 Differential Equations
- NCERT Exemplar Solutions for Class 12 Maths Chapter 9 Differential Equations
NCERT Solutions for Class 12 Mathematics: All Chapters
Chapter-by-chapter NCERT Solutions for the rest of Class 12 Mathematics, each mapped to the 2026-27 print.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1 | Relations and Functions NCERT Solutions |
| Chapter 2 | Inverse Trigonometric Functions NCERT Solutions |
| Chapter 3 | Matrices NCERT Solutions |
| Chapter 4 | Determinants NCERT Solutions |
| Chapter 5 | Continuity and Differentiability NCERT Solutions |
| Chapter 6 | Application of Derivatives NCERT Solutions |
| Chapter 7 | Integrals NCERT Solutions |
| Chapter 8 | Application of Integrals NCERT Solutions |
| Chapter 10 | Vector Algebra NCERT Solutions |
| Chapter 11 | Three Dimensional Geometry NCERT Solutions |
| Chapter 12 | Linear Programming NCERT Solutions |
| Chapter 13 | Probability NCERT Solutions |
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations: All Exercises
The table below lists every other exercise of Class 12 Maths Chapter 9 Differential Equations currently published, mapped to the 2026-27 NCERT.
| Exercise | NCERT Solutions |
|---|---|
| Exercise 9.1 | Exercise 9.1 Solutions (Order and Degree) |
| Exercise 9.3 | Exercise 9.3 Solutions (Variable Separable Method) |
| Exercise 9.4 | Exercise 9.4 Solutions (Homogeneous Differential Equations) |
| Exercise 9.5 | Exercise 9.5 Solutions (Linear Differential Equations) |
| Miscellaneous | Miscellaneous Exercise Solutions (Mixed-Method Problems) |
Differential Equations Class 12 NCERT Solutions: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
Exercise-wise Breakdown of the Differential Equations Chapter
The Differential Equations chapter splits into 5 numbered exercises plus a Miscellaneous Exercise. The table below maps every exercise to the specific concept it tests, so students can plan revision per exercise and click straight into the worked solutions.
| Exercise | Topic Tested |
|---|---|
| Exercise 9.1 | Basic concepts; order and degree of a differential equation |
| Exercise 9.2 | General and particular solutions |
| Exercise 9.3 | Formation of differential equation from family of curves |
| Exercise 9.4 | Variable separable differential equations |
| Exercise 9.5 | Homogeneous and linear differential equations |
| Miscellaneous Exercise | Mixed differential equations problems |
PDF Download Formats and Languages for the Differential Equations Chapter
The Differential Equations Class 12 PDF on this page is available in three formats - each suited to a different revision style. The table below summarises what each format is best for:
| Format | Best for | Approx. size |
|---|---|---|
| Normal-resolution PDF | Phone reading, quick revision between classes | 2-3 MB |
| HD PDF | Print-ready, desk study, board hall photocopy | 8-10 MB |
| Handwritten Notes PDF | Mirrors how a topper writes the chapter under Sunday-revision pace | 5-7 MB |
The differential equations class 12 ncert pdf and the parallel Hindi-medium edition both follow the same notation and equation numbering as the printed NCERT 2026-27 release. Key points students should know:
- NCERT-faithful: Every definition, theorem and exercise on the differential equations class 12 ncert pdf matches the printed textbook line for line.
- Hindi-medium edition: The differential equations class 12 pdf is also available in Hindi - same page numbering, same equation labels.
- Formula PDF separate: The differential equations class 12 formulas pdf is a one-page A4 reference sheet listing every identity used in the chapter.
- Solutions PDF separate: The differential equations class 12 solutions pdf gives every NCERT exercise worked out step by step.
- State-board alignment: Students on the Maharashtra board, HSC, or any state-board syllabus will find the same definitions in this differential equations class 12 pdf - only the exercise numbers differ.
Tip: Many toppers keep two parallel copies - a printed formula sheet on A4 for desk revision (the differential equations class 12 formulas pdf), and the full differential equations class 12 pdf on a phone for commute revision. Both files are free and linked above.
Important Questions and Previous Year Trends for the Differential Equations Chapter
The most repeated question patterns in CBSE Class 12 Maths for the Differential Equations chapter have settled into a stable cluster across 2019 to 2024 boards. Three question templates account for over 80% of the marks this chapter contributes:
| Template | Typical Marks | What it tests |
|---|---|---|
| Proof / property verification | 3 marks | Students show that a given relation/function/expression satisfies the chapter's definitions. |
| One-step computation | 2 marks | Substitution-based item: plug into a known formula and simplify. |
| Case-study scenario | 4 marks | Real-world setup applying the chapter's definitions, introduced in CBSE 2021+ papers. |
Walking through one example of each template before the exam covers most of the predictable differential equations class 12 important questions you will see on board day.
- differential equations class 12 previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
- The differential equations class 12 important questions with solutions set is reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching differential equations class 12 extra questions set adds advanced problems suitable for JEE Main and JEE Advanced.
- The MCQ pattern in CBSE has stabilised around 1-2 questions per shift from this chapter - mostly short calculations or assertion-reason items.
Year-wise PYQ Distribution
The table below maps the dominant question type asked from the Differential Equations chapter across recent CBSE Class 12 Maths boards:
| Year | Dominant Question Type | Approx. Marks |
|---|---|---|
| 2024 | Property verification + case-study item | 5-6 marks |
| 2023 | Computation with proof + assertion-reason MCQ | 5-6 marks |
| 2022 | Long-answer derivation + 2-mark substitution | 5-7 marks |
| 2021 | Definition recall + property check | 4-5 marks |
| 2020 | One-step computation + 3-mark proof | 5 marks |
The full differential equations class 12 important questions with solutions set (every year, every paper, every question type) is linked from the PYQ page at the bottom of this article.
How the Differential Equations Notes Pair with NCERT Solutions and the Formula Sheet
The Differential Equations Class 12 notes work best when paired with two sister resources from the Class 12 Maths hub. The table below shows how each resource fits into a typical revision week:
| Resource | Use it for | When |
|---|---|---|
| Differential Equations Notes (this page) | Theory, definitions, exam patterns | First pass, before practice |
| differential equations class 12 ncert solutions PDF | Step-by-step solved exercises | Second pass, during NCERT practice |
| differential equations class 12 formulas PDF | One-page identity recall | Third pass, alongside mock papers |
| Handwritten Notes PDF | Quick reading in topper's handwriting | Anytime, especially commute revision |
Around 60 percent of the chapter's scoring vocabulary appears on all three pages, so cross-resource use reinforces recall without adding study time.
- The differential equations class 12 ncert solutions cover every back-of-chapter exercise plus the miscellaneous exercise.
- The differential equations class 12 solutions for each individual exercise are indexed by exercise number on the sister NCERT Solutions page (see the Exercise-wise Breakdown table above for direct links).
- The differential equations class 12 formulas reference sheet is the same A4 file students sometimes refer to as differential equations class 12 all formulas - it lists every identity used in the chapter.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board differential equations class 12 textbook PDF all share the same core definitions.
- For class-first search phrasings - class 12 differential equations solutions, class 12 differential equations ncert solutions, ncert class 12 differential equations solutions - the same files cover the request.
Reference Books and State-Board Mapping
Students using reference books beyond NCERT, or studying under a state board, can map this chapter cleanly:
| Reference | How it maps to Differential Equations Class 12 |
|---|---|
| RD Sharma Class 12 Differential Equations | Question patterns overlap with NCERT at ~70%; an advanced supplement. |
| ML Aggarwal Class 12 Differential Equations | Solutions style is closer to JEE; good for problem-solving practice. |
| Teachoo differential equations class 12 | Free online walkthroughs; useful for video-style learning. |
| Shaalaa differential equations class 12 solutions | State-board (Maharashtra HSC) phrasings; same core definitions. |
| Maharashtra board differential equations class 12 textbook PDF | Same chapter content under the HSC syllabus; exercise numbers differ. |
| NCERT Exemplar Class 12 Differential Equations | Advanced problems for JEE Main/JEE Advanced preparation. |
All NCERT Solutions for Differential Equations Ex 9.2 with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 9 Differential Equations Ex 9.2 is listed below with its full Solution and Expert Solution hidden inside collapsible tabs. Click Check Solution to reveal the step-by-step working; click Expert Solution for the expanded explanation.
Questions
y = ex+1 : y'' - y' = 0.
y = x2+2x+C : y'-2x-2=0.
y = cos x + C : y' + sin x = 0.
y = √1+x2 : y' = xy1+x2.
y = Ax : xy' = y (x≠ 0).
y = xsin x : xy' = y + x√x2-y2 (x≠ 0 and x>y or x<-y).
xy = log y + C : y' = y21-xy (xy≠ 1).
y - cos y = x : (ysin y + cos y + x) y' = y.
x + y = tan-1y : y2y' + y2 + 1 = 0.
y = √a2-x2, x∈(-a,a) : x + y dydx = 0 (y≠ 0).
The number of arbitrary constants in the general solution of a differential equation of fourth order is: (A) 0 (B) 2 (C) 3 (D) 4.
The number of arbitrary constants in the particular solution of a differential equation of third order is: (A) 3 (B) 2 (C) 1 (D) 0.
How to Use the Differential Equations Notes Page Most Effectively
The recommended study plan for the Differential Equations Class 12 chapter splits across three sittings. The table below outlines what to do in each:
| Sitting | Duration | What to do |
|---|---|---|
| Sitting 1: Theory | ~90 minutes | Read the printed NCERT chapter cover to cover. Mark every definition and theorem statement. Then read the formula recall section on this page. |
| Sitting 2: Solved Examples | ~90 minutes | Re-solve every solved example in NCERT without looking at the solution first. Compare your steps against the printed working. Use the differential equations class 12 ncert solutions PDF if stuck. |
| Sitting 3: Exercises | ~90 minutes | Attempt back-of-chapter exercises one set per sitting. Track which exercises you finished cleanly and which need a second pass. Click into the linked exercise pages above for verification. |
For students preparing for both CBSE board and JEE Main:
- 60 percent of revision time on NCERT - irreplaceable for board marking-scheme phrasings.
- 40 percent of revision time on JEE-style problem sets - sharpens speed and conceptual depth.
- The differential equations class 12 important questions set on the previous-year page is the closest free analogue to a JEE-style problem set for this chapter.
- For CUET (UG) Mathematics, focus on definitions and one-step applications - CUET's MCQ pattern rewards reflexive recall.
Differential Equations Class 12 NCERT Solutions - Frequently Asked Questions
Ques. How many questions are in Class 12 Maths Chapter 9 Exercise 9.2?
Ans. Exercise 9.2 of Class 12 Maths Chapter 9 Differential Equations carries 12 questions in the 2026-27 NCERT. Q1 to Q10 ask you to verify that a given explicit or implicit function solves the corresponding differential equation, and Q11 to Q12 are single-correct MCQs on arbitrary constants.
Ques. How do you verify that a function is a solution of a differential equation in Class 12 Maths Chapter 9?
Ans. Differentiate the given function as many times as the order of the equation, substitute the function and its derivatives into the differential equation, and show the left-hand side reduces to the right-hand side for all values of x in the domain. If it does, the function is a solution.
Ques. What is the difference between general solution and particular solution in Class 12 Maths Chapter 9?
Ans. A general solution of an order-n differential equation contains n independent arbitrary constants. A particular solution is obtained by fixing every constant using given conditions, so it contains no arbitrary constants. This is exactly what Q11 and Q12 of Exercise 9.2 test.
Ques. How many arbitrary constants are in the general solution of a fourth-order differential equation?
Ans. Exactly four. The general solution of an order-n differential equation always contains n independent arbitrary constants, one introduced by each integration. This is Q11 of Exercise 9.2, and the correct option is (D) 4.
Ques. How do I download the Class 12 Maths Chapter 9 Exercise 9.2 NCERT Solutions PDF?
Ans. Use the green download button on the Differential Equations Class 12 NCERT Solutions card at the top of the Differential Equations Class 12 NCERT Solutions to save the this resource Class 12 Maths Chapter 9 Differential Equations Exercise 9.2 NCERT Solutions PDF. The file is free, ad-free, and mapped to the 2026-27 NCERT edition.
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