NCERT Solutions for Class 12 Maths Miscellaneous Exercise chapter 9 Differential Equations Miscellaneous

The Miscellaneous Exercise of NCERT Solutions for Class 12 Maths Chapter 9- Differential Equations consists of questions that covers all the topic in the chapter. Hence, the exercise is based on the following topics:

  1. Introduction to Differential Equations
  2. Basic Concepts of Differential Equation
  3. General and Particular Solutions of a Differential Equation
  4. Formation of a Differential Equation whose General Solution is given
  5. Methods of Solving First Order, First Degree Differential Equations

Download PDF of NCERT Solutions for Class 12 Maths Chapter 9- Differential Equations Miscellaneous Exercise

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Access Answers of Maths NCERT Class 12 Maths Chapter 9- Differential Equations Miscellaneous Exercise Page Number 419

1. For each of the differential equations given below, indicate its order and degree (if defined).

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 232

Solution:

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 233

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 234

Therefore, its degree is three.

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 235

2. For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 236

Solution:

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 237

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 238

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 239

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 240

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 241

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 242

3. Form the differential equation representing the family of curves given by (x – a)2 + 2y2 = a2, where a is an arbitrary constant.

Solution:

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 243

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 244

4. Prove that x2 – y2 = c (x2 + y2)2 is the general solution of differential equation (x3–3xy2) dx = (y3–3x2y) dy, where c is a parameter.

Solution:

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 245

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 246

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 247

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 248

5. Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.

Solution:

We know that the equation of a circle in the first quadrant with centre (a, a) and radius a which touches the coordinate axes is (x -a)2 + (y –a)2 = a2 …………1

Now differentiating above equation with respect to x, we get,

2(x-a) + 2(y-a) dy/dx = 0

⇒ (x – a) + (y – a) y’ = 0

On multiplying we get

⇒ x – a +yy’ – ay’ = 0

⇒ x + yy’ –a (1+y’) = 0

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 249

Therefore from above equation we have

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 250

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 251

6. Find the general solution of the differential equation

Solution:

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 252

On integrating, we get,

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 253

⇒ sin-1x + sin-1y = C

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 254

7. Show that the general solution of the differential equation

is given by (x + y + 1) = A (1 – x – y – 2xy), where A is parameter.

Solution:

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 255

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 256

8. Find the equation of the curve passing through the point (0, π/4) whose differential equation is sin x cos y dx + cos x sin y dy = 0.

Solution:

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 257

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 258

9. Find the particular solution of the differential equation (1 + e2x) dy + (1 + y2) ex dx = 0, given that y = 1 when x = 0.

Solution:

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 259

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 261
NCERT Solutions for Class 12 Maths Chapter 9 -  Image 260

10. Solve the differential equation

Solution:

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 262

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 263

11. Find a particular solution of the differential equation (x – y) (dx + dy) = dx – dy, given that y = –1, when x = 0. (Hint: put x – y = t)

Solution:

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 264

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 266
NCERT Solutions for Class 12 Maths Chapter 9 -  Image 265

12. Solve the differential equation

Solution:

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 267

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 268

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 269

13. Find a particular solution of the differential equation 

(x ≠ 0), given that y = 0 when x = π/2

Solution:

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 270

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 271

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 272

14. Find a particular solution of the differential equation,

given that y = 0 when x = 0.

Solution:

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 273

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 274

15. The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20, 000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?

Solution:

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 275

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 276

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 277

16. The general solution of the differential equation   is
A. xy = C B. x = Cy2 C. y = Cx D. y = Cx2

Solution:

C. y = Cx

Explanation:

Given question is

 
NCERT Solutions for Class 12 Maths Chapter 9 -  Image 278

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 279

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 280

17. The general solution of a differential equation of the type is

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 281

Solution:

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 282

Explanation:

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 283

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 284

18. The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
A. x ey + x2 = C B. x ey + y2 = C C. y ex + x2 = C D. y ey + x2 = C

Solution:

C. y ex + x2 = C

Explanation:

NCERT Solutions for Class 12 Maths Chapter 9 -  Image 285

Access other exercise solutions of Class 12 Maths Chapter 9- Differential Equations Miscellaneous Exercise

Exercise 9.1 Solutions 12 Questions

Exercise 9.2 Solutions 12 Questions

Exercise 9.3 Solutions 12 Questions

Exercise 9.4 Solutions 23 Questions

Exercise 9.5 Solutions 17 Questions

Exercise 9.6 Solutions 19 Questions

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