The Differential Equations Formulas list every formula, identity and theorem from Class 12 Mathematics Chapter 9 Differential Equations in a single PDF. The Differential Equations Formulas are arranged section by section, with each formula shown beside the conditions under which it is valid. The formula sheet is downloadable as a free file aligned to the 2026-27 NCERT syllabus.
- CBSE Weightage: 8 to 10 marks (typically one 5-mark long answer on the linear form plus a 3-mark variable-separation or homogeneous DE).
- JEE Main Weightage: 5 to 7% of the Maths section (2 to 3 questions per paper, order/degree plus one solved DE).
- CUET (UG) Weightage: 2 to 3 MCQs across most shifts, usually on order, degree, and the integrating factor.
Chapter 9 Differential Equations Formula Sheet PDF
This Formula Sheet is curated by Class 12 Maths experts at Collegedunia, mapped to the 2026-27 NCERT edition, and refined against the last five years of CBSE Board and JEE Main papers.
Why the Differential Equations Formula Sheet Matters for Class 12 Maths
The Differential Equations Formulas address this in the same order as the NCERT textbook.
Chapter 9 is the second-highest-marks calculus chapter in the CBSE Class 12 Maths paper after Integrals, and the single chapter where the solution method is fully formula-driven. Roughly 80% of CBSE board marks on Differential Equations are awarded for correctly identifying the standard form and applying its formula verbatim. Memorising the integrating factor recipe and the three standard forms is therefore the highest-use revision step before the exam.
Exam Tip: Every CBSE 5-mark Differential Equations question since 2019 has fallen into one of three buckets, variable separation, homogeneous, or linear in y. Identify the bucket in the first 30 seconds and the rest is mechanical.
Differential Equations Video Walkthrough
Source: Magnet Brains on YouTube
How will Collegedunia's Differential Equations Formula Sheet help You?
The Differential Equations Formulas address this in the same order as the NCERT textbook.
The Class 12 Maths Chapter 9 Differential Equations formula sheet from Collegedunia is built for the final 48-hour revision pass. It compresses 45 NCERT pages into six pages of pure formulas, organised by solution method so you can map any board paper question to the right template inside 30 seconds.
- Method-first layout orders formulas by solution technique (separation, homogeneous, linear), not by NCERT section, so you can pick the recipe straight off the question stem.
- Integrating factor recipe block isolates the four-step IF method as a standalone box, because it is the highest-marks formula in the Differential Equations Formulas.
- PYQ hit-rate tags flag which standard forms appeared in CBSE 2021 to 2025 and JEE Main 2021 to 2025 papers.
- Compact 6-page PDF that prints cleanly on three A4 sheets (double-sided) for the night-before revision.
Order and Degree of a Differential Equation: Class 12 Definitions
The Differential Equations Formulas address this in the same order as the NCERT textbook.
NCERT Section 9.2 defines the two classifying numbers attached to every DE. Order and degree decide which solution method applies; getting them wrong loses 1 mark on every multi-part CBSE question.
| Term | Definition | solved examples |
|---|---|---|
| Order | The order of the highest derivative appearing in the DE. | d2ydx2 + 3dydx = x has order 2. |
| Degree | The power of the highest-order derivative when the DE is polynomial in derivatives and free of radicals or fractions in them. | (dydx)3 + y = 0 has degree 3. |
| Degree not defined | When the DE cannot be written as a polynomial in its derivatives. | sin(dydx) + y = 0 has degree not defined. |
| Linear DE | Each term has degree at most 1 in y and its derivatives, with no products like y · y'. | dydx + Py = Q is linear. |
If the DE contains a fractional power of a derivative, rationalise it first; only then can degree be read off.
Variable Separable Differential Equations: The Separation Method
NCERT Section 9.4 introduces the simplest solvable form. Any DE that can be algebraically rearranged so that all y-terms sit on one side and all x-terms on the other is variable separable. Roughly one-third of CBSE 3-mark DE questions fall in this bucket.
Standard form:
dydx = f(x) g(y)
Separated form:
dyg(y) = f(x) dx
General solution:
∫ dyg(y) = ∫ f(x) dx + C
Three CBSE-recurring sub-patterns reduce to the separation method after a one-step rearrangement: dydx = ex+y , dydx = f(x)g(y) , and dydx = (1+x)(1+y2) . Each appeared at least twice in CBSE board papers between 2021 and 2025.
Homogeneous Differential Equations: The Substitution Method
| Step | Operation | Resulting Form |
|---|---|---|
| 1 | Verify dydx = F(yx) (degree-0 homogeneous). | dydx = F(yx) |
| 2 | Substitute y = vx ⇒ dydx = v + xdvdx . | v + xdvdx = F(v) |
| 3 | Separate the variables in v and x. | dvF(v) - v = dxx |
| 4 | Integrate both sides and back-substitute v = yx . | General solution in x, y. |
Quick Tip: If x and y appear symmetrically and the RHS depends only on x/y, the mirror substitution x = vy is faster than y = vx . CBSE marking schemes accept either path.
Linear Differential Equations and the Integrating Factor Formula
Linear form (in y):
dydx + P(x) y = Q(x)
Integrating factor:
IF = e∫ P(x) dx
General solution:
y · IF = ∫ Q(x)· IF dx + C
| Form | Integrating Factor | General Solution |
|---|---|---|
| dydx + Py = Q , P, Q functions of x | e∫ P dx | y · e∫ P dx = ∫ Q e∫ P dx dx + C |
| dxdy + P1x = Q1 , P1, Q1 functions of y | e∫ P1 dy | x · e∫ P1 dy = ∫ Q1 e∫ P1 dy dy + C |
Standard Forms of Differential Equations Tested in CBSE Class 12 Maths
| Standard Form | NCERT Section | Method | Key Formula |
|---|---|---|---|
| dydx = f(x) g(y) | 9.4 | Variable separation | ∫ dyg(y) = ∫ f(x) dx + C |
| dydx = F(yx) | 9.5 | Substitute y = vx | ∫ dvF(v) - v = ∫ dxx |
| dydx + P(x) y = Q(x) | 9.6 | Integrating factor | y · IF = ∫ Q · IF dx + C , IF = e∫ P dx |
| dxdy + P1(y) x = Q1(y) | 9.6 | Integrating factor (in y) | x · IF = ∫ Q1 · IF dy + C |
Formation of a Differential Equation from a Given Family of Curves
- One parameter family (e.g. y = mx ) gives a first-order DE after one differentiation.
- Two parameters (e.g. y = Asin x + Bcos x ) require two differentiations and yield a second-order DE.
- Standard CBSE question: form the DE for y = aex + be-x . Differentiating twice gives d2ydx2 = y .
Exam Tip: The order of the formed DE equals the number of arbitrary constants in the original curve family. CBSE 1-mark MCQs on this have appeared in three of the last five board papers.
Differential Equations Chapter Weightage Across CBSE, JEE Main and CUET
| Exam | 2025 | 2024 | 2023 | 2022 | 2021 |
|---|---|---|---|---|---|
| JEE Main (Maths) | 3 Qs | 2 Qs | 3 Qs | 2 Qs | 2 Qs |
| CBSE Class 12 Board | 9 marks | 10 marks | 8 marks | 9 marks | 8 marks |
| CUET (UG) Maths | 3 Qs | 2 Qs | 3 Qs | 2 Qs | - |
Most-Asked Differential Equations Formulas from CBSE Class 12 Board Papers
- IF = e∫ P dx for dydx + Py = Q - tested in CBSE 2021, 2022, 2024, 2025.
- y · IF = ∫ Q · IF dx + C (linear solution) - tested in CBSE 2022, 2023, 2025.
- Substitution y = vx for homogeneous DEs - tested in CBSE 2021, 2023, 2024.
- ∫ dyg(y) = ∫ f(x) dx + C (separation) - tested in CBSE 2022, 2023, 2025.
- Order plus degree identification on a given DE - tested in CBSE 2021, 2022, 2024 as a 1-mark MCQ.
Common Mistakes Students Make While Applying Differential Equations Formulas
- Forgetting the constant of integration + C after integrating the separated form, a guaranteed 1-mark deduction.
- Computing the integrating factor without the exponential: students sometimes write IF = ∫ P dx instead of e∫ P dx , losing the full method mark.
- Wrong identification of the linear form when P and Q are functions of y instead of x; the IF must then be e∫ P1 dy and the solution multiplies x, not y.
- Treating a homogeneous DE as separable without the y = vx substitution, leading to a non-integrable expression.
Common Mistake: The integrating factor for dydx + yx = x is e∫ 1x dx = eln x = x , not ln x . The exponential of the integral, not the integral itself, is the IF.
Related Resources for Class 12 Maths Chapter 9 Differential Equations
- NCERT Solutions 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
- Handwritten Notes for Class 12 Maths Chapter 9 Differential Equations
NCERT Formula Sheet for Class 12 Maths: All Chapters
| Chapter | Formula Sheet |
|---|---|
| Chapter 1 | Relations and Functions Formula Sheet |
| Chapter 2 | Inverse Trigonometric Functions Formula Sheet |
| Chapter 3 | Matrices Formula Sheet |
| Chapter 4 | Determinants Formula Sheet |
| Chapter 5 | Continuity and Differentiability Formula Sheet |
| Chapter 6 | Application of Derivatives Formula Sheet |
| Chapter 7 | Integrals Formula Sheet |
| Chapter 8 | Application of Integrals Formula Sheet |
| Chapter 10 | Vector Algebra Formula Sheet |
| Chapter 11 | Three Dimensional Geometry Formula Sheet |
| Chapter 12 | Linear Programming Formula Sheet |
| Chapter 13 | Probability Formula Sheet |
Exercise-wise Breakdown of the Differential Equations Chapter
| 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
| 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 |
- 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.
Important Questions and Previous Year Trends for the Differential Equations Chapter
| 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. |
- 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
| 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 |
How the Differential Equations Notes Pair with NCERT Solutions and the Formula Sheet
| 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 |
- 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
| 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. |
How to Use the Differential Equations Notes Page Most Effectively
| 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. |
- 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 Formulas - Frequently Asked Questions
Ques. How many formulas are there in Class 12 Maths Chapter 9 Differential Equations?
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