Chapter 9: Differential Equations Class 12 NCERT Solutions – Free Handwritten Notes Download in Hindi

Chapter 9: Differential Equations Class 12 NCERT Solutions – Free Handwritten Notes Download in Hindi: Differential Equations are mathematical equations that relate a function with its derivatives. These equations play a crucial role in understanding and modeling real-world dynamic systems where change occurs, such as in physics, biology, economics, and engineering. In this chapter, students are introduced to the basic concepts of differential equations, including the order and degree, and the process of forming differential equations from given functions.

The chapter focuses on solving first-order, first-degree differential equations using methods such as variable separable, homogeneous differential equations, and linear differential equations. These methods help students determine the original function from its derivative, a process known as solving the differential equation.

Students also explore general and particular solutions, and learn how to apply these in practical problems involving growth and decay, motion, and heat transfer. By understanding how to set up and solve differential equations, learners build a strong foundation for further studies in calculus and applied mathematics.

This chapter is not only significant from an academic perspective but also forms a basis for many entrance exams and higher studies in science and engineering. It enhances analytical thinking and problem-solving abilities.

Preview of Chapter 9: Differential Equations Class 12 Solutions Math’s Pdf | Free Handwritten Notes Download in Hindi

---

Images of Differential Equations Class 12 NCERT Solutions PDF

Related Free Notes from Studycart24.com of Class 12

• Class 12 Biology Handwritten Notes PDF Free Download- StudyCart24.com
• Class 12 Business Studies Notes: FREE Notes Chapter 1 to 12
• Complete Class 12 Chemistry Notes PDF with All Chapters for Free
• Class 12 Computer Science Notes PDF in English Free Download
• Class 12 Geography Notes: Download Free PDF of All Chapters in English
• Class 12 Physics Handwritten Notes in English PDF Download

---

Suggested Notes Links from Amazon Related to Class 12 Math’s Handwritten Notes

---

📘 Definition – Differential Equations (Class 12 Math’s Chapter 9)

A Differential Equation is an equation that involves a function and its derivatives.
It expresses the relationship between a dependent variable and one or more of its derivatives with respect to one or more independent variables.

✍️ Mathematically:

An equation of the form dydx=f(x,y)\frac{dy}{dx} = f(x, y)dxdy​=f(x,y)

or more generally, F(x,y,dydx,d2ydx2,… )=0F(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, \dots) = 0F(x,y,dxdy​,dx2d2y​,…)=0

is called a differential equation.

Key Features of Chapter 9: Differential Equations Class 12 Solutions Math’s Pdf | Free Handwritten Notes Download in Hindi

• Subject: Maths (Chapter 9: Differential Equations Class 12 )
• Language : Hindi
• Total pages : 114
• File size: 49.9 MB
• Format : PDF
• Well structured and easy to understand
• Includes importance formulas and definitions
• Covers all NCERT syllabus topics
• Useful for quick revision before exam

---

📘 10 Important Definitions – Chapter 9: Differential Equations (Class 12 Math’s)

1. Differential Equation:

An equation that involves derivatives of an unknown function is called a differential equation.
Example: dydx+y=ex\frac{dy}{dx} + y = e^xdxdy​+y=ex

---

2. Order:

The order of a differential equation is the highest order derivative present in the equation.
Example: d2ydx2+y=0\frac{d^2y}{dx^2} + y = 0dx2d2y​+y=0 ⇒ Order = 2

---

3. Degree:

The degree is the highest power of the highest order derivative, after the equation is made free from radicals and fractions.
Example: (d2ydx2)3+y=0(\frac{d^2y}{dx^2})^3 + y = 0(dx2d2y​)3+y=0 ⇒ Degree = 3

---

4. General Solution:

A general solution contains arbitrary constant(s) and represents a family of solutions.

---

5. Particular Solution:

A particular solution is obtained by giving specific values to the arbitrary constants in the general solution.

---

6. Formation of Differential Equation:

The process of eliminating arbitrary constants from a function to form a differential equation is called formation of differential equation.

---

7. Variable Separable Method:

A method of solving first-order differential equations where variables can be separated on opposite sides of the equation.

---

8. Homogeneous Differential Equation:

A differential equation of the form dydx=f(yx)\frac{dy}{dx} = f\left(\frac{y}{x}\right)dxdy​=f(xy​) is called homogeneous.

---

9. Linear Differential Equation:

An equation of the form dydx+Py=Q\frac{dy}{dx} + Py = Qdxdy​+Py=Q is called a linear differential equation, where PPP and QQQ are functions of xxx.

---

10. Integrating Factor (I.F.):

In a linear differential equation, an integrating factor is used to simplify and solve the equation. It is given by: I.F.=e∫PdxI.F. = e^{\int P dx}I.F.=e∫Pdx

---

📚 How to Prepare for Chapter 9: Differential Equations – Class 12 Math’s

✅ 1. Understand the Basics First

• Begin with the definition of a differential equation.
• Clearly understand order and degree with examples.
• Learn the difference between general and particular solutions.

---

✅ 2. Learn All Standard Forms

• First-order, first-degree equations.
• Variable separable method.
• Homogeneous differential equations.
• Linear differential equations (use of Integrating Factor).

---

✅ 3. Master the Methods of Solving

Type	Method Required
Variable separable	Separate variables and integrate
Homogeneous	Put y=vxy = vxy=vx, reduce and solve
Linear differential equation	Use Integrating Factor (I.F.)

---

✅ 4. Practice Step-by-Step Solving

• Solve NCERT examples and exercises completely.
• Focus on the method, not just the final answer.
• Write complete steps as required in board exams.

---

✅ 5. Memorize Key Formulas

• Integrating Factor: I.F.=e∫PdxI.F. = e^{\int P dx}I.F.=e∫Pdx
• Solution of linear DE: y(I.F.)=∫(Q⋅I.F.)dx+Cy(I.F.) = \int (Q \cdot I.F.) dx + Cy(I.F.)=∫(Q⋅I.F.)dx+C

---

✅ 6. Make a Formula + Concept Sheet

• Revise all types of forms and formulas regularly.
• Keep handwritten short notes for last-minute revision.

---

✅ 7. Solve Sample Papers & PYQs

• Practice previous year questions from CBSE and competitive exams (JEE, CUET).
• Solve sample papers within a time limit.

---

✅ 8. Avoid Common Mistakes

• Don’t forget to add the constant of integration (C).
• Carefully identify the type of differential equation before solving.
• Avoid algebraic mistakes while solving for variables.

---

✅ 9. Use Visual Aids if Needed

• Use graphs to understand growth/decay models.
• Try online apps for solving and checking DEs.

---

✅ 10. Stay Consistent with Practice

• Practice 2–3 questions daily.
• Don’t skip tough types—focus more on them.

---

📚 Chapters / Subtopics – Chapter 9: Differential Equations (Class 12 Math’s)

🔢 S.No.	📖 Subtopic Name	📝 What It Covers
1️⃣	Introduction to Differential Equations	Basic idea, real-life applications, and scope of the topic.
2️⃣	Basic Concepts	Definition of differential equation, order and degree, general and particular solutions.
3️⃣	Formation of Differential Equations	Forming DEs by eliminating arbitrary constants from given functions.
4️⃣	Methods of Solving First-Order, First-Degree DEs	Focus on equations like dydx=f(x,y)\frac{dy}{dx} = f(x, y)dxdy​=f(x,y)
5️⃣	Variable Separable Method	Solving DEs where variables can be separated (e.g., dydx=g(x)h(y)\frac{dy}{dx} = g(x)h(y)dxdy​=g(x)h(y)).
6️⃣	Homogeneous Differential Equations	Definition, substitution y=vxy = vxy=vx, and solving using variable separable method.
7️⃣	Linear Differential Equations	Solving equations of the form dydx+Py=Q\frac{dy}{dx} + Py = Qdxdy​+Py=Q using integrating factor.
8️⃣	Applications of Differential Equations (Optional/Introductory)	Basic problems based on growth, decay, motion, etc. (often covered conceptually).

---

📘 Why These Handwritten Notes Are Special for You?

(Chapter 9: Differential Equations – Class 12 Math’s)

🔢 No.	🌟 Feature/Benefit	📖 Description
1️⃣	Concept Clarity	Simplifies definitions, formulas, and solving methods step-by-step.
2️⃣	Board Exam Oriented	Focused on CBSE pattern with frequently asked questions and examples.
3️⃣	Easy to Revise	Well-structured layout ideal for quick revision and last-minute prep.
4️⃣	Visual Learning	Diagrams, underlining, and neat handwriting boost memory and understanding.
5️⃣	Complete Coverage	Includes order, degree, general/particular solutions, and all solving methods.
6️⃣	Solved Examples	Step-by-step solved examples build confidence and clarity.
7️⃣	Important Formulas Highlighted	All key formulas and tricks are boxed or underlined for easy spotting.
8️⃣	Compact & Time-Saving	No extra fluff — just what you need to score well.
9️⃣	Ideal for Self-Study	Perfect for independent learners — no tutor required.
🔟	Helpful for Competitive Exams	Builds base for JEE, CUET, NDA, and other entrance tests.

---

📚 Top 10 Benefits of Using Handwritten Notes

(Chapter 9: Differential Equations – Class 12 Maths)

🔢 No.	🌟 Benefit	📖 Why It Helps You
1️⃣	Better Concept Retention	Writing helps your brain understand and remember concepts more effectively.
2️⃣	Simplified Language	Notes are written in student-friendly terms with easy-to-follow steps.
3️⃣	Focus on Key Topics	Covers only what’s important – like order, degree, types of DEs, solving methods.
4️⃣	Quick Revision Tool	Neat structure helps revise complete chapter in less time before exams.
5️⃣	Boosts Visual Memory	Highlighted formulas, boxes, and arrows help in fast visual recall during exams.
6️⃣	Step-by-Step Solutions	Every method (like variable separable, linear DE) is shown in a structured format.
7️⃣	Saves Time While Studying	You don’t have to search through textbooks — all essentials are in one place.
8️⃣	Exam-Wise Practice	Includes board-style questions and solutions that match CBSE exam patterns.
9️⃣	Error-Free Understanding	Reduces confusion with clearly separated formulas, definitions, and solved examples.
🔟	Boosts Confidence & Speed	Regular revision from neat notes increases accuracy and reduces silly mistakes.

---

📘 Important Topics – Chapter 9: Differential Equations (Class 12 Math’s)

🔢 No.	📌 Topic	📖 What to Focus On
1️⃣	Definition of Differential Equation	Meaning, real-life use, notation of derivatives.
2️⃣	Order and Degree of a DE	How to identify order and degree with examples.
3️⃣	General and Particular Solutions	Difference between them and how to derive each.
4️⃣	Formation of Differential Equations	Elimination of arbitrary constants from a given function.
5️⃣	Variable Separable Method	Method of separating variables and integrating.
6️⃣	Homogeneous Differential Equations	Substitution y=vxy = vxy=vx or x=vyx = vyx=vy, converting and solving.
7️⃣	Linear Differential Equations	Standard form dydx+Py=Q\frac{dy}{dx} + Py = Qdxdy​+Py=Q, integrating factor, final solution.
8️⃣	Integrating Factor (I.F.)	How to find I.F. and apply it in linear equations.
9️⃣	Applications (Growth/Decay Models)	Simple word problems based on population, radioactive decay, etc.
🔟	NCERT Exercise Problems & Examples	Solve all examples & exercises thoroughly for board exam preparation.

---

🧮 Class 12 Math’s – Important Formulas

📘 Chapter 9: Differential Equations

🔢 No.	📌 Formula/Rule	📖 Description/Use
1️⃣	dydx=f(x)\frac{dy}{dx} = f(x)dxdy​=f(x)	Basic form of first-order differential equation
2️⃣	∫dydxdx=y=∫f(x)dx+C\int \frac{dy}{dx} dx = y = \int f(x) dx + C∫dxdy​dx=y=∫f(x)dx+C	General solution of separable differential equation
3️⃣	dydx=g(x)h(y)\frac{dy}{dx} = g(x)h(y)dxdy​=g(x)h(y)	Variable separable form
4️⃣	∫1h(y)dy=∫g(x)dx\int \frac{1}{h(y)} dy = \int g(x) dx∫h(y)1​dy=∫g(x)dx	Solve by separating variables and integrating
5️⃣	For homogeneous DE: y=vx⇒dydx=v+xdvdxy = vx \Rightarrow \frac{dy}{dx} = v + x \frac{dv}{dx}y=vx⇒dxdy​=v+xdxdv​	Used for solving homogeneous equations
6️⃣	Linear DE standard form: dydx+P(x)y=Q(x)\frac{dy}{dx} + P(x)y = Q(x)dxdy​+P(x)y=Q(x)	Recognizing a linear differential equation
7️⃣	Integrating Factor (I.F.): e∫P(x)dxe^{\int P(x) dx}e∫P(x)dx	Used to solve linear differential equations
8️⃣	Solution of linear DE: y(I.F.)=∫Q⋅I.F. dx+Cy(I.F.) = \int Q \cdot I.F. \, dx + Cy(I.F.)=∫Q⋅I.F.dx+C	General solution after multiplying by the integrating factor
9️⃣	General solution = Particular integral + Complementary function	Common in linear DEs (used in higher classes too)
🔟	Constant of integration: ‘+ C’	Always added after integrating a differential equation

---

📘 FAQs on Class 12 Math’s Handwritten Notes PDF Download (Differential Equations Class 12 NCERT Solutions)

---

📘 Class 12 Math’s Preparation Tips 🎯 Short & Effective Strategies

🔢 No.	✅ Tip	📖 How It Helps
1️⃣	Master NCERT First	Most board questions are directly from NCERT. Solve all examples and exercises.
2️⃣	Make a Formula Booklet	Write down all important formulas chapter-wise for quick revision.
3️⃣	Practice Daily – Even 30 Mins Helps	Consistency is key. 4–5 questions daily per chapter keeps your skills sharp.
4️⃣	Focus on Problem-Solving Techniques	Understand why a method is used, not just how.
5️⃣	Solve Previous Year Questions (PYQs)	Helps you spot question patterns and manage time better.
6️⃣	Mark Tough/Tricky Questions	So you can revisit them during revision.
7️⃣	Attempt Full-Length Sample Papers Weekly	Boosts exam confidence and improves speed + accuracy.
8️⃣	Avoid Mugging Up Formulas	Instead, understand their derivation and usage through examples.
9️⃣	Use Handwritten Notes	Easier to recall due to visual memory and simple presentation.
🔟	Revise Smart – Not Hard	Prioritize chapters with more weightage and scoring potential (like calculus).

---

⚠️ Avoid These Common Mistakes Chapter 9: Differential Equations – Class 12 Math’s

🔢 No.	❌ Common Mistake	✅ How to Avoid It
1️⃣	Confusing order with degree of a differential equation	Learn their definitions well; order = highest derivative, degree = its power
2️⃣	Forgetting to add the constant of integration (C)	Always check your final answer after integration
3️⃣	Not identifying the type of differential equation correctly	Before solving, classify as separable, linear, or homogeneous
4️⃣	Errors in separating variables	Carefully separate terms involving xxx and yyy before integrating
5️⃣	Incorrect use of Integrating Factor (I.F.) in linear DEs	Use correct formula: I.F.=e∫P(x)dxI.F. = e^{\int P(x) dx}I.F.=e∫P(x)dx
6️⃣	Skipping formation of differential equation steps	Always eliminate all constants properly and include all derivatives
7️⃣	Mistakes while substituting y=vxy = vxy=vx in homogeneous DEs	Differentiate using product rule: dydx=v+xdvdx\frac{dy}{dx} = v + x\frac{dv}{dx}dxdy​=v+xdxdv​
8️⃣	Leaving the solution without simplification	Final answers must be simplified and clearly written
9️⃣	Ignoring units in application-based problems (growth/decay)	Interpret real-world problems carefully; write correct expressions
🔟	Relying only on shortcuts without understanding methods	Practice full steps to avoid losing marks in step-marking scheme in CBSE exams

---

📘 Summary Table – Chapter 9: Differential Equations (Class 12 Math’s)

🔢 No.	📌 Topic	📖 Summary/Key Points
1️⃣	Differential Equation	Equation involving derivatives of a dependent variable with respect to independent ones.
2️⃣	Order	Highest order derivative present in the differential equation.
3️⃣	Degree	Highest power of the highest order derivative (after making it free from radicals/fractions).
4️⃣	General Solution	Contains arbitrary constants; represents a family of solutions.
5️⃣	Particular Solution	Obtained by assigning specific values to the constants in general solution.
6️⃣	Formation of DE	Done by eliminating arbitrary constants from a given function.
7️⃣	Variable Separable Method	Write equation as f(y)dy=g(x)dxf(y)dy = g(x)dxf(y)dy=g(x)dx and integrate both sides.
8️⃣	Homogeneous DE	Equation where dydx=f(yx)\frac{dy}{dx} = f\left(\frac{y}{x}\right)dxdy​=f(xy​), use substitution y=vxy = vxy=vx.
9️⃣	Linear DE	Of the form dydx+Py=Q\frac{dy}{dx} + Py = Qdxdy​+Py=Q, solve using Integrating Factor.
🔟	Integrating Factor (I.F.)	I.F.=e∫PdxI.F. = e^{\int P dx}I.F.=e∫Pdx; used to solve linear differential equations.

---

✅ Useful for:

• CBSE Board Exams
• JEE Mains (Basic Level)
• CUET/Entrance Revision

---

✅ Conclusion – Chapter 9: Differential Equations (Class 12 Math’s)

Chapter 9, Differential Equations, is one of the most important and application-based chapters in Class 12 Mathematics. It lays the foundation for advanced studies in calculus, physics, engineering, and mathematical modelling. This chapter teaches students how to deal with equations that involve derivatives — a tool for understanding change in various systems.

Students learn key concepts such as order, degree, and the difference between general and particular solutions. They also master methods to solve first-order, first-degree differential equations, including variable separable, homogeneous, and linear differential equations using integrating factors.

Another critical aspect covered is the formation of differential equations by eliminating arbitrary constants — a skill useful in deriving models from real-life data.

By the end of this chapter, students are equipped not only with solving techniques but also with a deeper understanding of how mathematical relationships describe real-world phenomena like population growth, radioactive decay, and motion.

With consistent practice and conceptual clarity, students can score well in board exams and build a strong base for competitive exams. Overall, this chapter enhances problem-solving ability, mathematical reasoning, and real-world application skills.

Tags:-
Chapter 9 Differential Equations Class 12
Class 12 Maths Chapter 9 Notes
Differential Equations Class 12 PDF
NCERT Solutions Differential Equations Class 12
Class 12 Maths Handwritten Notes Chapter 9
Differential Equations Important Formulas Class 12
Class 12 Maths Chapter 9 Solutions
CBSE Class 12 Differential Equations
Linear Differential Equations Class 12
Homogeneous Differential Equations Class 12
Variable Separable Method Class 12
Formation of Differential Equations Class 12
General and Particular Solution Class 12
Integrating Factor Class 12
Differential Equation Solving Methods Class 12
Class 12 Maths Chapter 9 Summary
Differential Equations Class 12 Important Questions
CBSE Board Maths Chapter 9 Preparation
NCERT Chapter 9 Class 12 Maths PDF Download
Maths Class 12 Chapter 9 Revision Notes
Class 12 Maths Chapter 9 Mind Map
Class 12 Maths Chapter-wise Notes PDF
Chapter 9 Differential Equations Practice Questions
Differential Equations Class 12 Worksheet
Class 12 Chapter 9 Differential Equations Derivations
Differential Equations Class 12 NCERT Solutions