The Differential Equations Class 12 NCERT Solutions for Class 12 Mathematics Chapter 9 Differential Equations Exercise 9.3 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.
| Exercise 9.3 snapshot | Detail |
|---|---|
| Total problems | 23 (Q1 to Q23) |
| Question split | Q1 to Q10 general solution, Q11 to Q14 particular solution, Q15 to Q22 word problems, Q23 MCQ |
| Core method | Separate to h(y) dy = g(x) dx , then integrate both sides |
The split above shows why Exercise 9.3 is long but predictable: the same separation step opens every problem, and only the integral on each side changes.
- Question count: 23 problems in total, the largest practice block before the homogeneous and linear methods.
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 first rearranges the equation to the form h(y) dy = g(x) dx , then integrates each side with the standard integral named explicitly. The half-angle identity in Q1, the sin-1 form in Q2, partial fractions in Q11 and Q12, and the exponential-growth template dNdt = kN in Q20 to Q22 are all written in full.
The Collegedunia editorial team has matched every general and particular solution against the official NCERT answer key and the 2026-27 textbook, including the numeric answers for the balloon, compound-interest and bacteria problems.
Why Exercise 9.3 Matters Most in Class 12 Maths Chapter 9 Differential Equations
The Differential Equations Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
The variable-separable method is the foundation the homogeneous and linear methods build on, and it is the only technique the application word problems use. JEE Main has carried a separable-method question in 4 of the last 5 papers, almost always disguised as a growth or decay model identical in structure to Q20 to Q22.
Differential Equations Ex 9 3 Video Walkthrough
Source: Magnet Brains on YouTube
How Collegedunia's NCERT Solutions Help You Clear Exercise 9.3
The Differential Equations Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
Exercise 9.3 is long, and the marks are lost in the integration step, not the separation step. The Collegedunia solutions name every standard integral before applying it, so the working is examination-ready.
- Separation shown first in every problem, so the structure is visible before any integration.
- Standard integrals named, such as ∫ dy√a2-y2 = sin-1ya for Q2.
- Partial fractions solved fully for Q11 ( (x+1)(x2+1) ) and Q12 ( x(x-1)(x+1) ).
- Initial condition applied last in Q11 to Q14, after the general solution, to fix the constant.
- Word-problem modelling spelled out for the balloon (Q19), compound interest (Q20, Q21) and bacteria (Q22).
Differential Equations Class 12 NCERT Solutions Exercise 9.3: Question-Wise Answer Map
The Differential Equations Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
The table lists the headline answer for representative problems across the four blocks of Exercise 9.3. Use it to verify your own working after attempting the set.
| Q No. | Equation / model | Answer |
|---|---|---|
| 1 | dydx = 1-cos x1+cos x | y = 2tan(x/2) - x + C |
| 2 | dydx = √4-y2 | y = 2sin(x+C) |
| 4 | sec2xtan y dx + sec2ytan x dy = 0 | tan x tan y = C |
| 6 | dydx = (1+x2)(1+y2) | tan-1y = x + x33 + C |
| 11 | (x3+x2+x+1)y' = 2x2+x , y(0)=1 | y = 12log|x+1| + 34log(x2+1) - 12tan-1x + 1 |
| 14 | dydx = ytan x , y(0)=1 | y = sec x |
| 17 | Tangent-slope times y equals x, through (0,-2) | y2 - x2 = 4 |
| 19 | Balloon, volume changes at constant rate | r(t) = [3]63t+27 |
| 20 | Rs 100 doubles in 10 years, continuous | r ≈ 6.93% per year |
| 22 | Bacteria up 10% in 2 hours, reach 2,00,000 | t = 2log 2log 1.1 ≈ 14.55 hours |
| 23 | MCQ: general solution of dydx = ex+y | (A) ex+e-y=C |
The four word problems Q19 to Q22 carry the same exponential or power-law structure, so learn the dNdt = kN template clears all of them at once. These application problems are the 5-mark Long Answer source from this exercise.
Variable Separable Method: The Three-Step Routine for Class 12 Maths Exercise 9.3
Every problem in Exercise 9.3 follows the same routine. Internalising it makes the long set fast.
Step 2. Integrate both sides, naming the standard integral used on each side and adding a single arbitrary constant.
Step 3. If an initial condition is given, substitute it into the general solution to fix the constant and obtain the particular solution.
For word problems, an extra step precedes Step 1: translate the verbal statement into a differential equation, such as "rate proportional to count" becoming dNdt = kN .
Common Mistakes Students Make in Class 12 Maths Exercise 9.3
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.
- Applying the initial condition before finding the general solution, instead of after.
- Forgetting partial fractions in Q11 and Q12 and trying to integrate the rational function directly.
- Missing the f'(x)f(x) pattern in Q5 and Q10, which integrates to log|f(x)| .
- In word problems, treating "constant rate of change of volume" as constant rate of change of radius.
CBSE Board Exam Relevance of Class 12 Maths Chapter 9 Exercise 9.3
Separable-method questions appear as 2, 3 and 5 mark items, with the application word problems being the recurring Long Answer. The table maps the last five CBSE sittings.
| Year | Marks from Ex 9.3 style | Question tested |
|---|---|---|
| 2025 | 5 | Bacterial growth model, find time to reach a target population |
| 2024 | 3 | General solution of a separable equation with partial fractions |
| 2023 | 3 | Particular solution from an initial condition |
| 2022 | 5 | Continuous compound-interest rate problem |
| 2021 | 2 | General solution of dydx = ex+y type equation |
Exercise 9.3 style content has carried 2 to 5 marks every single year for the last five sittings, making it the highest-yield exercise of Chapter 9 for CBSE.
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 9.2 | Exercise 9.2 Solutions (Verify Given Function is a Solution) |
| 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 9.3 with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 9 Differential Equations Ex 9.3 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
dydx = 1-cos x1+cos x.
dydx = √4-y2 (-2
dydx + y = 1 (y≠ 1).
sec2x tan y dx + sec2y tan x dy = 0.
(ex+e-x) dy - (ex-e-x) dx = 0.
dydx = (1+x2)(1+y2).
ylog y dx - x dy = 0.
x5dydx = -y5.
dydx = sin-1x.
extan y dx + (1-ex)sec2y dy = 0.
(x3 + x2 + x + 1)dydx = 2x2 + x; y=1 when x=0.
x(x2-1)dydx = 1; y=0 when x=2.
cos(dydx) = a (a∈R); y=1 when x=0.
dydx = ytan x; y=1 when x=0.
Find the equation of a curve passing through the point (0,0) whose differential equation is y' = exsin x.
For the DE xydydx = (x+2)(y+2), find the solution curve passing through (1,-1).
Find the equation of a curve passing through (0,-2) given that at any point (x,y) on the curve, the product of the slope of its tangent and the y-coordinate equals the x-coordinate.
At any point (x,y) on a curve, the slope of the tangent is twice the slope of the line segment joining (x,y) to (-4,-3). The curve passes through (-2,1). Find the equation.
The volume of a spherical balloon being inflated changes at a constant rate. Initially the radius is 3 units; after 3 seconds the radius is 6 units. Find the radius after t seconds.
In a bank, the principal increases continuously at r% per year. Find r if Rs. 100 doubles itself in 10 years (given e2 = 0.6931).
In a bank, the principal increases continuously at 5% per year. Rs. 1000 is deposited; how much will it be worth after 10 years? (Given e0.5=1.648.)
In a culture, the bacteria count is 1,00,000. The number rises by 10% in 2 hours. In how many hours will the count reach 2,00,000, given that the growth rate is proportional to the count?
The general solution of the differential equation dy/dx = ex+y is: (A) ex+e-y=C (B) ex+ey=C (C) e-x+ey=C (D) e-x+e-y=C.
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 9.3?
Ans. Exercise 9.3 of Class 12 Maths Chapter 9 Differential Equations carries 23 questions in the 2026-27 NCERT. Q1 to Q10 ask for the general solution, Q11 to Q14 for a particular solution, Q15 to Q22 are word problems, and Q23 is a single-correct MCQ.
Ques. What is the variable separable method in Class 12 Maths Chapter 9?
Ans. The variable separable method solves an equation that can be written as h(y) dy = g(x) dx . You move every y term to one side with dy and every x term to the other with dx, then integrate both sides and add a single arbitrary constant.
Ques. How do you solve growth and decay word problems in Class 12 Maths Exercise 9.3?
Ans. Translate "rate proportional to the quantity" into dNdt = kN , separate to dNN = k dt , integrate to N = N0ekt , then use the given data points to compute k and the required time or quantity. Q20, Q21 and Q22 use this template.
Ques. What is the general solution of dy/dx = e^(x+y)?
Ans. Write ex+y = exey , separate to e-y dy = ex dx , and integrate to get ex + e-y = C . This is Q23 of Exercise 9.3, and the correct option is (A).
Ques. How do I download the Class 12 Maths Chapter 9 Exercise 9.3 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 9.3 NCERT Solutions PDF. The file is free, ad-free, and mapped to the 2026-27 NCERT edition.
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