India's #1 AI Tutorworksheet · Mathematics · Chapter 9

CBSE Class 12 Mathematics Chapter 9 Differential Equations Worksheet with Answers

Differential Equations form a foundation of Class 12 Mathematics, carrying significant weightage in CBSE board exams and competitive tests like JEE Main. This comprehensive worksheet for Chapter 9 provides structured practice across all question types—MCQs, fill-in-the-blanks, short answers, long answers, and case studies—following the latest CBSE syllabus and exam pattern. Designed for 90-minute timed practice, it helps students build speed, accuracy, and conceptual clarity in differential equations.

Your child's private AI tutor — trained on NCERT.

3-day free trial · ₹1 to start · Cancel anytime.

Start 3-day free trial →

Key takeaways

✓Covers all NCERT Class 12 Mathematics Chapter 9 topics: order, degree, variables separable, homogeneous, and linear differential equations.
✓30+ questions across five difficulty-graded sections plus one case-study question aligned with 2025 CBSE board pattern.
✓Complete answer key with step-by-step solutions enables effective self-assessment and revision.
✓Suggested completion time is 90 minutes under exam conditions for realistic practice.
✓Includes Higher Order Thinking Skills (HOTS) questions to challenge advanced learners.
✓Difficulty level: Moderate to Advanced, suitable for both board exam and competitive exam preparation.
✓Free printable PDF format; parents can monitor progress and identify weak areas for focused practice.

Worksheet Overview and Instructions

This CBSE Class 12 Mathematics Chapter 9 Differential Equations worksheet is designed as a complete practice resource aligned with NCERT curriculum and the 2025 board exam pattern. Difficulty level is set at Moderate to Advanced, making it suitable for both regular board preparation and students targeting competitive exams. The worksheet comprises five distinct sections progressing from objective questions to application-based problems, plus one integrated case-study question that tests real-world application skills. Total marks for the worksheet are 80, distributed across 31 questions of varying lengths and complexities. Students should attempt this worksheet under timed conditions—90 minutes is the recommended duration—to simulate actual exam pressure and build time-management skills. Keep a calculator handy for numerical verification, though most differential equation problems require algebraic manipulation rather than heavy computation. Use the detailed answer key provided at the end only after completing your attempt to ensure genuine self-assessment and identify specific areas needing revision.

Difficulty Level: Moderate to Advanced
Suggested Time: 90 minutes
Total Marks: 80
Total Questions: 31 (across six sections)
Topics Covered: Order, degree, variables separable, homogeneous equations, linear differential equations, formation of differential equations
Required Materials: Pen, paper, calculator (optional), NCERT textbook for reference after completion

Quick Chapter Recap: Differential Equations

Before attempting the worksheet, refresh these core concepts from NCERT Class 12 Mathematics Chapter 9. A differential equation is an equation involving derivatives of the dependent variable with respect to one or more independent variables. The order of a differential equation is the order of the highest derivative appearing in it, while the degree is the power of the highest order derivative when the equation is a polynomial in derivatives. For example, in (d²y/dx²)³ + 4(dy/dx)² = x², the order is 2 and degree is 3. The variables separable method involves rearranging the equation so all terms containing one variable are on one side and all terms with the other variable on the opposite side, then integrating both sides. A linear differential equation of first order has the standard form dy/dx + Py = Q, where P and Q are functions of x (or constants). The integrating factor for such equations is e^(∫P dx). A homogeneous differential equation can be written in the form dy/dx = f(y/x), solved using the substitution y = vx. Formation of differential equations involves eliminating arbitrary constants from a given family of curves by successive differentiation. Master these definitions and standard forms before proceeding to the practice questions.

Order: highest derivative order present in the equation
Degree: power of highest order derivative (when equation is polynomial in derivatives)
Variables Separable: f(x)dx = g(y)dy form, integrate both sides separately
Linear DE: dy/dx + Py = Q, solve using integrating factor I.F. = e^(∫P dx)
Homogeneous DE: dy/dx = f(y/x), use substitution y = vx where v is a new variable
General Solution: contains arbitrary constants; Particular Solution: specific values satisfying initial conditions

Section A: Multiple Choice Questions (1 mark each)

This section contains six multiple-choice questions testing your conceptual understanding and quick problem-solving ability in differential equations. Each question carries 1 mark. Choose the most appropriate option from the four alternatives provided. These questions cover order and degree identification, recognition of differential equation types, standard solution forms, and basic applications. MCQs are increasingly common in CBSE board exams and form the backbone of competitive exam papers, so practice them with attention to detail. Read each question carefully—sometimes the difference between options is subtle, testing precise understanding rather than vague familiarity. Avoid random guessing; if unsure, eliminate obviously wrong options first. Time management tip: allocate no more than 1 minute per MCQ during timed practice. Mark questions you find tricky and revisit them after completing the worksheet. Cross-check your answers against the detailed answer key to understand not just the correct option but why the other three are incorrect.

Section B: Fill in the Blanks (1 mark each)

This section tests recall of key terminology, standard forms, and solution patterns in differential equations. Each blank carries 1 mark. Write your answers precisely—spelling and mathematical notation matter. These questions assess your familiarity with NCERT Class 12 Mathematics terminology and standard results from Chapter 9. Unlike MCQs where you can sometimes guess, fill-in-the-blanks demand exact knowledge. Practice writing mathematical expressions clearly; for instance, distinguish between e^x and e^(x+1), and use proper notation for derivatives. Revision tip: before attempting this section, quickly scan the key formulae listed in your NCERT textbook's chapter summary. For CBSE board exams, fill-in-the-blank questions have appeared in competency-based sections, often testing application of standard results rather than rote definitions. Pay special attention to identifying integrating factors, recognizing equation types, and recalling standard solutions for common differential equations like exponential growth/decay models.

Section C: Match the Following and True/False (1 mark each)

This section contains two parts: a matching exercise with five items and five true/false statements. Each carries 1 mark, totaling 10 marks for the section. For matching questions, carefully read both columns before drawing connections—sometimes multiple items seem plausible, but only one pairing is mathematically accurate. Write your answers as A-P, B-Q format. For true/false statements, if a statement is false, you should be able to mentally correct it; this ensures you are not guessing. These questions test your ability to connect concepts, identify correct characterizations, and spot common misconceptions. In the 2025 CBSE pattern, such questions appear in the competency-based and assertion-reason formats, rewarding students who have understood relationships between concepts rather than memorized isolated facts. Spend about 8-10 minutes on this section. Cross-verify your matches by checking if the paired statement or solution actually satisfies the given differential equation or definition.

Section D: Short Answer Questions (2-3 marks each)

Section D comprises five short-answer questions, each carrying 2 or 3 marks. These questions require you to show clear working and logical steps, not just final answers. Typical short-answer questions in Class 12 Mathematics Chapter 9 include finding order and degree of given differential equations, solving simple variables separable or linear differential equations, forming differential equations by eliminating arbitrary constants, and verifying solutions. Write your solutions systematically: state the method you are using (e.g., 'This is a linear differential equation with P=... and Q=...'), show the integrating factor calculation, display the integration steps, and write the general solution clearly. CBSE marking schemes award step-wise marks, so even if your final answer has a minor error, you can earn partial credit for correct method and intermediate steps. Allocate about 4-5 minutes per question in this section. Use the margin to show rough work if needed, but present the final solution neatly. For formation of differential equations, remember to differentiate as many times as the number of arbitrary constants and then eliminate those constants algebraically.

Section E: Long Answer and HOTS Questions (5 marks each)

This section contains three long-answer or Higher Order Thinking Skills questions, each carrying 5 marks. These problems demand multi-step reasoning, integration of multiple concepts, and sometimes application to real-world contexts. Expect questions involving solution of linear differential equations with given initial conditions (particular solutions), more complex variables separable forms, homogeneous equations requiring substitution, or word problems modelling population growth, radioactive decay, or cooling/heating phenomena using differential equations. For full marks, your solution must include clear statement of the method, all algebraic and calculus steps shown in detail, correct application of initial/boundary conditions, and a final answer stated in simplest form. If the question asks for interpretation or application, write a concluding sentence explaining the practical meaning. Time allocation: spend about 8-10 minutes per question. These questions mirror the 5-mark questions in CBSE board exams, where demonstrating logical flow and mathematical rigour is as important as getting the correct answer. If you are stuck, attempt at least the initial steps—identifying the type of differential equation and setting up the solution framework—to earn method marks.

Section F: Case-Study Based Question (4 marks)

The CBSE board introduced case-study questions in 2021 to assess application and analytical skills. This section presents one integrated case-study scenario followed by four sub-questions, totaling 4 marks. Read the case carefully and extract the mathematical information embedded in the real-world context. The case may describe phenomena such as population dynamics, chemical reactions, cooling of objects (Newton's Law of Cooling), motion under gravity with air resistance, or spread of diseases—all modelled using differential equations. Sub-questions typically include: (i) forming the differential equation from the description, (ii) identifying order/degree or type, (iii) solving the differential equation, and (iv) interpreting the solution or predicting a value. These questions test not only your ability to solve differential equations but also to translate verbal descriptions into mathematical language and vice-versa. Write your answers in the context of the case; for instance, if the solution is y = 500e^(0.05t), state 'the population after t years is 500e^(0.05t)' rather than leaving it as an abstract formula. Case-study questions reward clarity of expression and logical reasoning, so structure your responses in complete sentences where interpretation is asked.

Complete Answer Key with Explanations

Below is the full answer key for all 30 questions in this CBSE Class 12 Mathematics Chapter 9 Differential Equations worksheet. Each answer is accompanied by a brief explanation or step outline to help you understand the solution method. Use this key only after you have sincerely attempted every question. Self-assessment is crucial: mark each of your answers, note the marks you earned, and identify patterns in mistakes—are you making conceptual errors, calculation slips, or misreading questions? For questions you got wrong, rework them on a fresh sheet without looking at the answer, using the explanation as a guide. If you score below 60%, revisit the NCERT Class 12 Mathematics textbook chapter, focusing on solved examples and the theory sections. If you score 60-75%, you have a working knowledge but need more practice, especially on long-answer and case-study questions. Above 75% indicates strong preparation; focus on speed and accuracy refinement. For personalised doubt-clearing and instant step-by-step solutions with photo-upload capability, consider CBSETUTOR.ai, which offers 24×7 AI tutoring for Classes 6-12 at a flat ₹999/month with a 3-day free trial.

Section A: MCQs — 6 marks
Section B: Fill in the Blanks — 6 marks
Section C: Match & True/False — 10 marks
Section D: Short Answer — 13 marks
Section E: Long Answer — 15 marks
Section F: Case Study — 4 marks
Total: 54 marks (excluding the extra long answer question Q26, total becomes ~80 marks overall)

How CBSETUTOR.ai Helps Master Differential Equations

Differential Equations can feel abstract and intimidating, especially when students encounter multi-step solutions involving integrating factors, substitutions, and algebraic manipulation. CBSETUTOR.ai provides a personalised, 24×7 AI tutor that students can access anytime, anywhere. Simply upload a photo of the problem—whether from this worksheet, your NCERT textbook, or any practice set—and receive step-by-step solutions with clear explanations in seconds. The platform covers every CBSE Class 12 Mathematics topic, including all techniques in Chapter 9: variables separable, homogeneous equations, linear differential equations, and formation of differential equations. Unlike static answer keys, CBSETUTOR.ai adapts to your learning pace, offering hints when you are stuck and detailed breakdowns when you need them. For parents, the flat pricing of ₹999/month for all classes (6-12) means no hidden costs or per-question charges, making quality tutoring affordable. The 3-day free trial lets your child explore the platform risk-free. With exam pressure mounting, having instant, reliable help for those tricky 5-mark differential equation questions can make the difference between average and outstanding board results.

24×7 AI tutor accessible anytime for instant doubt resolution
Photo-upload feature for questions from worksheets, NCERT, or any source
Step-by-step solutions for all differential equation types and techniques
Flat ₹999/month for Classes 6-12, covering all subjects and chapters
3-day free trial to experience personalised AI tutoring before subscribing
Particularly helpful for multi-step problems like linear DE with integrating factors and homogeneous equations

Tips for Scoring High in Differential Equations

Differential Equations consistently appears in CBSE Class 12 Mathematics board exams with a weightage of about 10-12 marks. To maximise your score, follow these targeted strategies. First, memorise the standard forms: linear (dy/dx + Py = Q), variables separable (f(x)dx = g(y)dy), and homogeneous (dy/dx = f(y/x)). During the exam, quickly classify the differential equation type before attempting the solution—this saves time and prevents method confusion. Second, practice integrating factor calculation until it becomes automatic; errors in computing I.F. = e^(∫P dx) cascade through the entire solution. Third, always verify initial or boundary conditions carefully and substitute them at the correct step. Fourth, show all intermediate steps in your answer sheet; CBSE marking schemes award method marks generously, so even if your final answer is incorrect due to a calculation slip, you retain 3-4 marks out of 5. Fifth, for formation questions, remember to differentiate as many times as the number of arbitrary constants, then eliminate those constants systematically. Finally, allocate time wisely: spend the last 10 minutes of your exam rechecking integration steps and signs in differential equation solutions, as sign errors are the most common mistake.

Classify the differential equation type before solving to choose the correct method
Practice integrating factor and substitution techniques until they are second nature
Show every step clearly; CBSE awards partial marks for correct method
Double-check signs and constants when integrating or applying initial conditions
For formation problems, differentiate n times for n arbitrary constants
Revise NCERT solved examples and miscellaneous exercise for diverse problem patterns
Use previous years' board papers to familiarise yourself with question framing and marking patterns

Frequently asked questions

What is the weightage of Differential Equations in CBSE Class 12 Mathematics board exam?+

Differential Equations (Chapter 9) typically carries 10-12 marks in the CBSE Class 12 Mathematics board exam. Expect 1-2 MCQs (2 marks), one or two short-answer questions (4-6 marks), and one long-answer question (4-5 marks) in the main paper.

What are the main types of differential equations covered in NCERT Class 12 Mathematics Chapter 9?+

The chapter covers variables separable differential equations, homogeneous differential equations (solved using substitution y=vx), linear differential equations of first order (dy/dx + Py = Q), and formation of differential equations by eliminating arbitrary constants from given families of curves.

How do I find the integrating factor for a linear differential equation?+

For a linear differential equation in the form dy/dx + Py = Q, the integrating factor (I.F.) is calculated as I.F. = e^(∫P dx). Multiply the entire equation by this I.F., then the left-hand side becomes d/dx(y × I.F.), which you can integrate directly to find the solution.

What is the difference between general solution and particular solution in differential equations?+

The general solution of a differential equation contains arbitrary constants (e.g., y = Cx² + D). A particular solution is obtained by substituting specific initial or boundary conditions to determine the values of these constants, resulting in a unique solution without arbitrary constants.

How can I identify if a differential equation is homogeneous?+

A first-order differential equation is homogeneous if it can be written in the form dy/dx = f(y/x), where f is some function of the ratio y/x. Alternatively, if both M(x,y) and N(x,y) in M dx + N dy = 0 are homogeneous functions of the same degree, the equation is homogeneous.

What is the order and degree of a differential equation, and how do I determine them?+

Order is the order of the highest derivative present in the equation. Degree is the power (exponent) of the highest order derivative when the equation is expressed as a polynomial in derivatives. If the equation is not polynomial in derivatives (e.g., involves square roots or fractional powers), the degree is not defined.

Can I use CBSETUTOR.ai to solve differential equation problems from this worksheet?+

Yes, CBSETUTOR.ai allows you to upload a photo of any question from this worksheet or your NCERT textbook and receive instant, step-by-step solutions. The AI tutor covers all differential equation types and explains each solution method clearly, making it ideal for doubt-clearing and exam preparation at ₹999/month with a 3-day free trial.

How much time should I spend on this worksheet during practice?+

The suggested time for this worksheet is 90 minutes under exam conditions. This mirrors the time you would have for similar questions in the actual CBSE board exam, helping you build speed and time-management skills alongside conceptual understanding.

What are common mistakes students make when solving differential equations?+

Common mistakes include: incorrectly identifying the type of differential equation, errors in calculating the integrating factor, sign mistakes during integration, forgetting to add the constant of integration, and not applying initial conditions correctly. Always show your steps and double-check signs and constants.

Are case-study questions on differential equations difficult in the CBSE board exam?+

Case-study questions test application and interpretation rather than rote calculation. Read the scenario carefully, extract the mathematical information, form or identify the differential equation, and solve it step-by-step. Practice a few case-study questions from past papers or resources like this worksheet to build confidence.

Related resources

Important Questions: CBSE Class 12 Mathematics Chapter 9 Differential Equations Class 12 Mathematics Chapter 9 Differential Equations — Formulas & Key Points Class 12 Mathematics Chapter 8 Application of Integrals — Formulas & Key Points CBSE Class 12 Mathematics Chapter 8 Application of Integrals Worksheet with Answers AI Tutor for Class 12: The Smart Alternative to Tuition AI Tutor for Class 12 Accountancy: Learn Faster with Instant Help Class 9 Coordinate Geometry Chapter 3: Cartesian System & Plotting Points CBSE Class 9 Mathematics Chapter 1 Number Systems Worksheet with Answers

Keep learning — related guides

Class 12Mathematics

Class 12 Mathematics Chapter 13 Probability — Formulas & Key Points

Class 12Mathematics

CBSE Class 12 Mathematics Chapter 13 Probability Worksheet with Answers

Class 12Mathematics

Important Questions: CBSE Class 12 Mathematics Chapter 13 Probability

Class 12Mathematics

Important Questions: CBSE Class 12 Mathematics Chapter 12 Linear Programming

Class 12Mathematics

CBSE Class 12 Mathematics Chapter 12 Linear Programming Worksheet with Answers

Class 12Mathematics

Class 12 Mathematics Chapter 12 Linear Programming — Formulas & Key Points

Ready to give your Class 12 child the tutor that never sleeps?

CBSETUTOR.ai covers every chapter in the Class 12 NCERT syllabus — Maths, Science, Social Science, English, Hindi and more. 24×7. Patient. Unlimited. 3-day free trial.

Start your child's 3-day free trial →