The Differential Equations Class 12 NCERT Solutions for Class 12 Mathematics Chapter 9 Differential Equations Miscellaneous Exercise 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.
| Miscellaneous Exercise snapshot | Detail |
|---|---|
| Total problems | 18 (Q1 to Q18) |
| Methods blended | Order/degree, verification, separable, homogeneous, linear (IF), initial conditions |
| MCQ count | 3 (Q16, Q17, Q18) |
The snapshot shows why this exercise is the Differential Equations Class 12 NCERT Solutions's exam rehearsal: every technique appears once, and the first decision in each problem is which method to apply.
- Question count: 18 problems with no method label, so the strategy choice is half the work.
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 opens by naming the method the equation calls for, then executes it end to end: order-and-degree for Q1, four verifications in Q2, a homogeneous proof in Q3, separable integration in Q5 and Q6, and integrating-factor work in Q11 and Q12. The Collegedunia editorial team has matched every answer against the official NCERT answer key and the 2026-27 textbook.
Why the Miscellaneous Exercise Decides Your Class 12 Maths Chapter 9 Score
The Differential Equations Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
In the CBSE board paper and JEE Main, differential-equation questions never carry a method tag, so the skill the Miscellaneous Exercise builds, recognising the form first, is the one that is actually examined. JEE Main has lifted a Miscellaneous-style mixed problem in 4 of the last 5 papers. This exercise is the closest practice to the real test.
Differential Equations Misc Video Walkthrough
Source: Magnet Brains on YouTube
How Collegedunia's NCERT Solutions Help You Clear the Miscellaneous Exercise
The Differential Equations Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
Marks in this exercise are lost at the decision step, before any algebra. The Collegedunia solutions state the method-recognition reasoning before solving, which is the habit that transfers to the exam.
- Method named first in every problem, with the structural signal that points to it.
- Substitution hints honoured, such as x - y = t in Q10 and the dxdy linear form in Q12.
- Verification problems in Q2 solved as four independent identity checks.
- Proof-type problems Q3 and Q5 shown as full derivations, not just answer statements.
- Initial conditions applied last in Q8, Q11, Q13 and Q14 to fix the constant.
Differential Equations Class 12 NCERT Solutions Miscellaneous Exercise: 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 Miscellaneous Exercise. Use it to verify your own working after attempting the set.
| Q No. | Task / equation | Answer |
|---|---|---|
| 1 | Order and degree of three equations | (2,1), (1,3), (4, not defined) |
| 2 | Verify four given functions solve their equations | All four verified |
| 3 | Prove x2-y2=c(x2+y2)2 solves the given DE | Proven |
| 4 | dydx + √1-y2√1-x2 = 0 | sin-1x + sin-1y = C |
| 6 | Curve through (0,π4) , sin xcos y dx + cos xsin y dy = 0 | √2cos xcos y = 1 |
| 7 | (1+e2x) dy + (1+y2)ex dx = 0 , y(0)=1 | tan-1y + tan-1ex = π2 |
| 8 | yex/y dx = (xex/y+y2) dy | ex/y = y + C |
| 11 | dydx + ycot x = 4xcsc x , y(π/2)=0 | ysin x = 2x2 - π22 |
| 12 | (x+1)dydx = 2e-y - 1 , y(0)=0 | (2-ey)(x+1) = 1 |
| 16 | MCQ: general solution of y dx - x dyy = 0 | (C) y = Cx |
| 18 | MCQ: general solution of ex dy + (yex+2x) dx = 0 | (C) yex + x2 = C |
Q3 and Q5 are proof-type problems where the working, not just the final relation, earns the marks, which is why they map to the Differential Equations Class 12 NCERT Solutions's 5-mark Long Answer. Q11 and Q12 are the recurring particular-solution items in CBSE board papers.
Method-Recognition Checklist for the Class 12 Maths Chapter 9 Miscellaneous Exercise
Use this checklist as the first decision on any unlabelled differential equation.
2. Is every term the same total degree in x and y? It is homogeneous; substitute y = vx .
3. Is it of the form dydx + Py = Q (or the x-in-y sister form)? Use the integrating factor e∫ P dx .
4. If a substitution hint is given (such as x-y=t in Q10), apply it before classifying.
This four-question filter clears every problem in the Miscellaneous Exercise and is the same filter the JEE and CBSE papers expect you to run silently.
Common Mistakes Students Make in the Class 12 Maths Chapter 9 Miscellaneous Exercise
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.
- Skipping the order-and-degree justification in Q1 and writing the degree without the polynomial-in-derivatives reason.
- Treating a proof-type problem (Q3, Q5) as answer-only and losing the derivation marks.
- Ignoring the substitution hint in Q10, which makes the equation needlessly hard.
- Applying initial conditions before the general solution is complete in Q7, Q11 and Q12.
CBSE Board Exam Relevance of Class 12 Maths Chapter 9 Miscellaneous Exercise
The Miscellaneous Exercise is the direct source of the chapter's 3 and 5 mark questions because both papers test method recognition. The table maps the last five CBSE sittings.
| Year | Marks from Misc style | Question tested |
|---|---|---|
| 2025 | 5 | Particular solution of a linear equation with a trig coefficient |
| 2024 | 5 | Proof that a given relation is the general solution of a stated DE |
| 2023 | 3 | Separable equation with an inverse-trig answer |
| 2022 | 3 | Homogeneous equation with an exponential substitution |
| 2021 | 3 | Particular solution from an initial condition |
The Miscellaneous Exercise has fed 3 to 5 marks of the Differential Equations block every single year for the last five sittings, the highest concentration of method-mix marks in Chapter 9 for this resource learners.
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 9.3 | 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) |
NCERT Solutions Class 12 Maths: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
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 Misc with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 9 Differential Equations Misc 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
Indicate the order and degree (if defined) of each of the following differential equations.
(i) d2ydx2 + 5x(dydx)2 - 6y = log x.
(ii) (dydx)3 - 4(dydx)2 + 7y = sin x.
(iii) d4ydx4 - sin(d3ydx3) = 0.
For each verify that the given function (implicit or explicit) is a solution of the corresponding DE.
(i) xy = aex + be-x + x2 : xd2ydx2 + 2dydx - xy + x2 - 2 = 0.
(ii) y = ex(acos x + bsin x) : d2ydx2 - 2dydx + 2y = 0.
(iii) y = xsin 3x : d2ydx2 + 9y - 6cos 3x = 0.
(iv) x2 = 2y2log y : (x2+y2)dydx - xy = 0.
Prove that x2 - y2 = c(x2+y2)2 is the general solution of (x3 - 3xy2) dx = (y3 - 3x2y) dy, where c is a parameter.
Find the general solution of the differential equation dydx + √1-y2√1-x2 = 0.
Show that the general solution of dydx + y2+y+1x2+x+1 = 0 is (x+y+1) = A(1 - x - y - 2xy), where A is a parameter.
Find the equation of the curve passing through (0,π4) whose differential equation is sin xcos y dx + cos xsin y dy = 0.
Find the particular solution of (1+e2x) dy + (1+y2)ex dx = 0, given y = 1 when x = 0.
Solve y ex/y dx = (xex/y + y2)dy (y≠ 0).
Find a particular solution of (x-y)(dx + dy) = dx - dy, given y = -1 when x = 0.
Hint: put x-y = t.
Solve (e-2√x√x - y√x)dxdy = 1 (x≠ 0).
Find a particular solution of dydx + ycot x = 4xcsc x (x≠ 0), given y = 0 when x = π2.
Find a particular solution of (x+1)dydx = 2e-y - 1, given y = 0 when x = 0.
The general solution of y dx - x dyy = 0 is: (A) xy = C (B) x = Cy2 (C) y = Cx (D) y = Cx2.
The general solution of an equation of the form dxdy + P1x = Q1 is:
(A) y e∫ P1 dy = ∫ (Q1e∫ P1 dy) dy + C
(B) y· e∫ P1 dx = ∫ (Q1e∫ P1 dx) dx + C
(C) x· e∫ P1 dy = ∫ (Q1e∫ P1 dy) dy + C
(D) x· e∫ P1 dx = ∫ (Q1e∫ P1 dx) dx + C.
The general solution of ex dy + (yex + 2x) dx = 0 is: (A) xey + x2 = C (B) xey + y2 = C (C) yex + x2 = C (D) yey + x2 = 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 the Class 12 Maths Chapter 9 Miscellaneous Exercise?
Ans. The Miscellaneous Exercise of Class 12 Maths Chapter 9 Differential Equations carries 18 questions in the 2026-27 NCERT. It blends order-and-degree, verification, separable, homogeneous and linear problems, with Q16 to Q18 being single-correct MCQs.
Ques. Why is the Miscellaneous Exercise the most important set in Class 12 Maths Chapter 9?
Ans. Because no problem carries a method label, it forces you to recognise the form first, exactly as the CBSE and JEE papers do. This recognition skill, not the algebra, is what the exam actually tests, which makes the Miscellaneous Exercise the closest practice to the real paper.
Ques. How do you decide which method to use on an unlabelled differential equation?
Ans. Run a four-step filter: try variable separation first, then check homogeneity by degree, then check the linear standard form for the integrating factor, and finally apply any given substitution hint. The this resource solutions show this decision for every Miscellaneous problem.
Ques. What is the general solution of (y dx - x dy)/y = 0?
Ans. Rearranging gives dxx = dyy , which integrates to log x = log y + log C , so y = Cx . This is Q16 of the Miscellaneous Exercise, and the correct option is (C).
Ques. How do I download the Class 12 Maths Chapter 9 Miscellaneous Exercise NCERT Solutions PDF?
Ans. Use the green download button on the NCERT Solutions Class 12 Maths card at the top of the Differential Equations Class 12 NCERT Solutions to save the this resource Class 12 Maths Chapter 9 Differential Equations Miscellaneous Exercise NCERT Solutions PDF. The file is free, ad-free, and mapped to the 2026-27 NCERT edition.
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