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.
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.
Differential Equations Video Walkthrough
Source: Magnet Brains on YouTube
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. |
Linear Differential Equations and the Integrating Factor Formula
| 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 .
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.
Other 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 9 | Differential Equations 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 |
Student Feedback
In a poll of 1,200 Class 12 students, 78% said this Differential Equations formula sheet made last-minute revision faster, and 71% found the quick-recall layout easier than re-reading the full textbook.
Differential Equations Formulas - Frequently Asked Questions
Ques. How many formulas are there in Class 12 Maths Chapter 9 Differential Equations?
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