NCERT Solutions Class 12 Maths Chapter 9 guides you through Differential Equations with detailed solutions to all 98 questions across 6 exercises. You’ll learn how to form differential equations, solve first-order equations using variable separation and integrating factors, find general and particular solutions, and apply these techniques to real-world problems in physics and engineering. Each solution includes the exact method examiners expect, with shortcuts for solving homogeneous and linear differential equations quickly.
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All exercises with step-by-step solutions | Updated 2025-26 | Free Download
Download PDF (Free)NCERT Solutions Class 12 Maths Chapter 9 Differential Equations – Complete Guide
NCERT Class 12 Chapter 9 on Differential Equations introduces you to one of the most powerful mathematical tools used in physics, engineering, economics, and biological sciences. You’ll explore how rates of change are expressed mathematically and learn systematic methods to solve equations involving derivatives. This chapter carries significant weightage of 7 marks in the CBSE board exam and is considered challenging yet highly rewarding for students aiming for top scores.
📊 CBSE Class 12 Maths Chapter 9 – Exam Weightage & Marking Scheme
| CBSE Board Marks | 7 Marks |
| Unit Name | Calculus |
| Difficulty Level | Hard |
| Importance | High |
| Exam Types | CBSE Board, State Boards |
| Typical Questions | 2-3 questions |
You will begin by understanding the basic concepts of differential equations, their order, and degree. The chapter then progresses to practical methods of solving first-order, first-degree differential equations through variable separable forms, homogeneous equations, and linear differential equations. You’ll also learn how to form differential equations from given general solutions and apply these concepts to solve real-world problems involving growth and decay, Newton’s law of cooling, and motion problems.
For CBSE board exams, expect a mix of 2-mark questions on basic concepts and identification, 4-mark questions on solving specific types of differential equations, and a crucial 6-mark problem-solving question. The chapter requires strong algebraic manipulation skills and a thorough understanding of integration techniques from Chapter 7. Practice is essential as questions often involve multiple steps and careful attention to initial conditions.
Quick Facts – Class 12 Chapter 9
| 📖 Chapter Number | Chapter 9 |
| 📚 Chapter Name | Differential Equations |
| ✏️ Total Exercises | 6 Exercises |
| ❓ Total Questions | 98 Questions |
| 📅 Updated For | CBSE Session 2025-26 |
Mastering differential equations not only helps you score well in boards but also builds a foundation for advanced mathematics and engineering courses. The logical approach to problem-solving and the ability to model real situations mathematically are skills that extend far beyond examinations, making this chapter one of the most practically relevant topics in your Class 12 mathematics curriculum.
NCERT Solutions Class 12 Maths Chapter 9 – All Exercises PDF Download
Download exercise-wise NCERT Solutions PDFs for offline study
| Exercise No. | Topics Covered | Download PDF |
|---|---|---|
| Exercise 9.1 | Complete step-by-step solutions for 12 questions | 📥 Download PDF |
| Exercise 9.2 | Complete step-by-step solutions for 12 questions | 📥 Download PDF |
| Exercise 9.3 | Complete step-by-step solutions for 23 questions | 📥 Download PDF |
| Exercise 9.4 | Complete step-by-step solutions for 17 questions | 📥 Download PDF |
| Exercise 9.5 | Complete step-by-step solutions for 19 questions | 📥 Download PDF |
| Miscellaneous Exercise on Chapter 9 | Complete step-by-step solutions for 15 questions | 📥 Download PDF |
Differential Equations – Key Formulas & Concepts
Quick reference for CBSE exams
| Formula | Description | When to Use |
|---|---|---|
| Order of a Differential Equation Order = Highest order derivative present | The order is the highest derivative that appears in the equation. Note: Look for \( \frac{d^ny}{dx^n} \), the highest ‘n’ is the order. | Identifying the order of a given differential equation. |
| Degree of a Differential Equation Degree = Power of the highest order derivative (after removing radicals and fractions) | The degree is the power to which the highest order derivative is raised, *after* the equation is made free of radicals and fractions in derivatives. Note: Make sure the equation is a polynomial equation in derivatives first! If not, the degree is not defined. | Identifying the degree of a given differential equation. |
| General Solution Solution containing arbitrary constants | The general solution includes arbitrary constants and represents a family of solutions. Note: The number of arbitrary constants equals the order of the differential equation. | Finding the most general form of the solution to a differential equation, typically involves integration. |
| Particular Solution Solution obtained from the general solution by giving particular values to the arbitrary constants | The particular solution is a specific solution obtained by applying initial or boundary conditions to the general solution. Note: First find the general solution, then use the initial conditions to solve for the constants. | Finding a specific solution when initial conditions (like y(0) = 2) are given. |
| Variable Separable Form \(f(x)dx = g(y)dy\) | A differential equation where terms involving only x can be separated from terms involving only y. Note: Integrate both sides separately after separating the variables. | Solving differential equations where you can isolate x and y terms on opposite sides. |
| Homogeneous Differential Equation \(\frac{dy}{dx} = F(\frac{y}{x})\) | A differential equation where \( \frac{dy}{dx} \) can be expressed as a function of \( \frac{y}{x} \) only. Note: Substitute \( y = vx \), then \( \frac{dy}{dx} = v + x\frac{dv}{dx} \). This transforms the equation into variable separable form. | Recognizing and solving homogeneous differential equations. Substitute \( y = vx \) to solve. |
| Linear Differential Equation of the Form dy/dx + Py = Q \(\frac{dy}{dx} + Py = Q\) | A first-order linear differential equation where P and Q are functions of x only. Note: First find the integrating factor (IF). Then, the solution is \( y(IF) = \int Q(IF) dx + C \) | Solving first-order linear differential equations. |
| Integrating Factor (IF) for dy/dx + Py = Q \(IF = e^{\int P dx}\) | The integrating factor helps solve linear differential equations. Note: Remember to integrate P with respect to x. The constant of integration is not needed when finding IF. | When solving a linear differential equation of the form \( \frac{dy}{dx} + Py = Q \). |
| Solution of dy/dx + Py = Q \(y \cdot IF = \int Q \cdot IF \ dx + C\) | The general solution to the linear differential equation. Note: Don’t forget the constant of integration, C. | After finding the integrating factor for a linear differential equation. |
| Linear Differential Equation of the Form dx/dy + Px = Q \(\frac{dx}{dy} + Px = Q\) | A first-order linear differential equation where P and Q are functions of y only. Note: First find the integrating factor (IF). Then, the solution is \( x(IF) = \int Q(IF) dy + C \). Note that the integration is with respect to *y*. | Solving first-order linear differential equations where x is the dependent variable and y is the independent variable. |
| Integrating Factor (IF) for dx/dy + Px = Q \(IF = e^{\int P dy}\) | The integrating factor helps solve linear differential equations. Note: Remember to integrate P with respect to y. The constant of integration is not needed when finding IF. | When solving a linear differential equation of the form \( \frac{dx}{dy} + Px = Q \). |
| Solution of dx/dy + Px = Q \(x \cdot IF = \int Q \cdot IF \ dy + C\) | The general solution to the linear differential equation. Note: Don’t forget the constant of integration, C. Note that the integration is with respect to *y*. | After finding the integrating factor for a linear differential equation where x is the dependent variable. |
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