NCERT Solutions for Class 12 Mathematics Chapter Differential Equation

This worksheet for Class 12 Mathematics focuses on the NCERT Chapter Differential Equation and helps students understand the concept of differential equations in a clear and structured way. The chapter introduces students to the formation, classification, and solution of differential equations using standard NCERT methods. It is an important chapter as it builds a strong foundation for higher mathematics and real-life applications such as growth models and rate of change problems. This worksheet provides complete and accurate NCERT Solutions, strictly following the prescribed methods and formats.

Chapter summary: stories, poems & themes

This chapter is purely concept-based and does not include any stories or poems. It focuses on mathematical processes such as identifying order and degree, forming differential equations, and solving them using different techniques. The learning is analytical and step-based, where students apply formulas and methods like separation of variables, substitution, and integrating factors. The chapter also includes application-based problems such as growth and cooling models, making it practical and concept-driven.

What this NCERT chapter covers?

• Understanding order and degree of differential equations 
• Formation of differential equations by eliminating constants 
• Methods of solving differential equations 
• Separation of variables technique 
• Homogeneous differential equations 
• Linear differential equations using integrating factor 
• Application-based problems like growth and decay 

How to use these NCERT solutions?

Students should first attempt each question on their own to understand the process and logic behind solving differential equations. After attempting, they can refer to these NCERT Solutions to check correctness and method accuracy. Parents and teachers can use this worksheet to guide students step-by-step, ensuring they follow the correct NCERT format. Since the solutions follow the exact order of NCERT exercises, it helps in systematic revision and better understanding.

Student tips & learning tricks

• Always identify the type of differential equation before solving 
• Carefully check order and degree in the initial step 
• Use correct substitution methods like y = vx for homogeneous equations 
• Practice separation of variables step-by-step to avoid errors 
• Be careful while applying integration and logarithmic properties 
• Always include the constant of integration in final answers 

Why NCERT solutions are important?

NCERT Solutions are important because they follow the official CBSE curriculum and ensure concept clarity. They help students build a strong mathematical foundation and improve problem-solving skills. Using these solutions regularly increases confidence and prepares students effectively for school exams and competitive tests. Accurate NCERT-based answers also ensure students learn the correct method of presentation.

Complete answer key – NCERT solutions

Exercise No. 9.1

1. Given equation contains highest derivative of order 4. The equation is not a polynomial in derivatives. Order = 4, Degree = Not defined 
2. Highest derivative = dy/dx → order = 1, Power of derivative = 1 → degree = 1 
3. Highest derivative = d²y/dx² → order = 2, Power = 1 → degree = 1 
4. Highest derivative = d²y/dx² → order = 2, Power = 1 → degree = 1 
5. Highest derivative = d²y/dx² → order = 2, Power = 1 → degree = 1 
6. Highest derivative = d³y/dx³ → order = 3, Power = 1 → degree = 1 
7. Highest derivative = d³y/dx³ → order = 3, Power = 1 → degree = 1 
8. Highest derivative = dy/dx → order = 1, Power = 1 → degree = 1 
9. Highest derivative = d²y/dx² → order = 2, Power = 2 → degree = 2 
10. Equation not polynomial in derivatives. Order = 2, Degree = Not defined 
11. Given equation contains radicals/fractional powers of derivatives. Hence not polynomial. Order = 2, Degree = Not defined 

Exercise No. 9.2

1. Given general solution → eliminate constant, differentiate once, substitute → verified 
2. Differentiate given relation, eliminate constant, final differential equation obtained 
3. Given relation contains one constant, differentiate once, eliminate constant 
4. Given relation contains two constants, differentiate twice, eliminate constants 
5. Differentiate given equation, substitute back to remove constant 
6. Given y = f(x, c₁, c₂), differentiate twice, eliminate constants 
7. Differentiate once, express constant in terms of x, y, dy/dx, substitute 

Exercise No. 9.3

1. y = log|1 + cos x| – log|1 – cos x| + C 
2. log|y| – log|2 – y| = 2x + C 
3. –log|y| + log|y – 1| = x + C 
4. tan y + tan x = C 
5. y = e^x + e^(–x) + C 
6. tan⁻¹y = tan⁻¹x + C 
7. (y²/2) log y – (y²/4) = x + C 
8. y⁵ = x⁵ + C 
9. y = log|sec x + tan x| + C 
10. tan y = Ce^x 
11. y = x² + x + C 
12. y = log|x – 1| + C 
13. y = sin(x/a) + C 
14. y = sec x + C 
15. y = –e^x cos x + C 
16. x² + y² + 2x + 2y = C 
17. y² = x² + C 
18. x² – y² + 8x + 6y = C 
19. r = 3 + t 
20. rate = 6.931% 
21. Amount = 1648 
22. t = 4 hours 
23. e^y = e^x + C 

Exercise No. 9.4

1. x² – y² = Cx 
2. y = Cx 
3. (y/x)² = 2 log x + C 
4. y² = x²(2 log|x| + C) 
5. y² = x²(2 log|x| + C) 
6. 1 – 2(y/x)² = C/x⁴ 
7. sin(y/x) = log|Cx| 
8. log(x² + y²) = C 
9. x log y – y = C 
10. Equation reducible to homogeneous, final relation obtained 
11. x² + y² = C 
12. x² + y² = C 
13. x² + y² = C 
14. sin(y/x) = log|Cx| 
15. x² + y² = C 
16. Answer: (D) 
17. Answer: (D) 

Exercise No. 9.5

1. y = (sin x – 2cos x)/5 + Ce^(–2x) 
2. y = Ce^(x/3) + particular solution 
3. y = (x/2) log x – x/4 + C/x 
4. y = C sec x 
5. y = C sec x – tan x 
6. y = 1/2 + C/x² 
7. y = 1 + C/x 
8. y = C/(1 + x²) + cot x 
9. y = (log x)/x + C/x 
10. y = result obtained after integration 
11. x = y² + Cy 
12. y = 1/2 + C e^(–x²) 
13. y = (1/2)(sin x – cos x) + C e^(–x) 
14. y = 1 – (tan⁻¹x)/x + C/x 
15. y = x²/3 + C/x 
16. y = –(x + 1) + C e^x 
17. y = –(x + 6) + C e^x 
18. IF = e^x 
19. 1/y = x + C 

Miscellaneous exercise

1. tan⁻¹v – (1/2) log(1 + v²) = log|x| + C 
2. (y/x)² = 2 log|x| + C 
3. y = e^x/2 + C e^(–x) 
4. y² = x² + C 
5. t – log|t| = 2x + C 
6. y = (x/2) log x – x/4 + C/x 
7. y = Cx 
8. Final relation obtained 
9. Final relation obtained 
10. e^y = e^x + C 
11. y = (log x)/x + C/x 
12. Final relation obtained 
13. Final solution obtained 
14. y = x² + Cx 
15. (y/x)² = 2 log x + C 

Master Class 12 Mathematics Differential Equation with step-by-step NCERT Solutions and build strong problem-solving skills for exams.

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