CBSE Class 12 Differential Equations PYQs Analysis (2021-2025)

Based on detailed analysis of last 5 years' papers. Perfect for 2026 Boards prep!

Key Trends

Weightage Breakdown: 1-2 MCQs/assertion-reason (1 mark), 1-2 short answers (2-3 marks), 1 long answer on solving DE (4-6 marks), occasional case-based.
Repeated Topics: Order & degree of DE, formation of DE from family of curves, variable separable, homogeneous equations, linear first-order DE (integrating factor), Bernoulli equation (rare), orthogonal trajectories (occasional), applications (exponential growth/decay, Newton's law of cooling).
Difficulty: Medium; identifying type & correct method (separable vs homogeneous vs linear) is key. Integrating factor for linear DE is most repeated.
Changes: More focus on solving linear DE and application-based questions in 2023–2025 papers.

Most Important and Repeated Questions

Question Example	Type/Marks	Years Repeated	Notes
Solve the differential equation: dy/dx = (x + y + 1)/(x + y - 1) or similar homogeneous type.	Long Answer (4-5 marks)	2021, 2022, 2023, 2024, 2025	Repeated 5x; Substitute v = x + y → becomes separable; log form solution.
Solve the linear differential equation: dy/dx + Py = Q (e.g., dy/dx + y cot x = 2x sin x or dy/dx + y/x = sin x).	Long Answer (4-6 marks)	2021 Term 2, 2022, 2023, 2024	Repeated 4x; Integrating factor = e^{∫P dx}; multiply & integrate.
Form the differential equation of family of curves y = Ae^{kx} + Be^{-kx} or y = a cos(x + b) or circle family.	Short Answer (3 marks)	2022, 2023, 2024, 2025	Repeated 4x; Eliminate arbitrary constants by differentiating & combining equations.
Assertion: The general solution of a first-order DE has one arbitrary constant. Reason: First-order DE has one parameter family of solutions.	Assertion-Reason (1 mark)	2023, 2024, 2025	Repeated 3x; Both true, reason explains.
Solve separable equation: dy/dx = y/x or (1+x²)dy/dx = 1+y² or similar.	Short Answer (2-3 marks)	2021 Term 1, 2022, 2023, 2025	Repeated 4x; Separate variables → integrate both sides → log or arctan form.
In a culture, bacteria grow at rate proportional to number present. If initially 500, after 2 hrs 1600, find after 4 hrs (exponential growth).	Application (3-4 marks)	2022, 2023, 2024	Repeated 3x; dy/dt = ky → y = y₀ e^{kt}; find k, then y(4).
MCQ: The degree of the differential equation [1 + (dy/dx)²]^{3/2} = d²y/dx² is: (a) 1 (b) 2 (c) 3 (d) not defined	MCQ (1 mark)	2021 Term 1, 2023, 2024	Repeated 3x; Answer (b) 2 (highest power of highest derivative after clearing).
Solve Bernoulli equation: dy/dx + Py = Q y^n (rare but appeared as variant of linear).	Short Answer (3 marks)	2023, 2025	Repeated 2x; Substitute v = y^{1-n} → linear in v.
Case-based: Form DE from given geometrical condition or solve given DE with initial condition.	Case-Based (4 marks)	2023, 2025	Repeated 2x; Apply method based on type (homogeneous/linear/separable).
Find orthogonal trajectories of family y = ce^x or x² + y² = c² (circle family).	Short Answer (3 marks)	2021 Term 2, 2024	Repeated 2x; Replace dy/dx by -dx/dy in DE of family → solve.

Best Format for Preparation and Answering

Study Format: Classify DE type quickly (separable/homogeneous/linear). Memorize integrating factors for linear (e^{∫P dx}). Practice 10–12 PYQs daily, especially linear & homogeneous.
Answer Format: 1. Identify type & method, 2. Show substitution (if homogeneous v=y/x or Bernoulli v=y^{1-n}), 3. Integrate step-by-step, 4. Write general + particular solution (with initial condition if given).
Tips: Linear first-order DE (with IF) and homogeneous are highest repeat & marks. Always verify solution by differentiation if time permits. Time: 10–12 mins for 4–6 mark questions. Very scoring chapter once methods are clear.

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