NCERT Solutions for Class 12 Maths Chapter 9 - Differential Equations Exercise 9.3

NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations Exercise 9.3 – CBSE Free PDF Download

Exercise 9.3 of NCERT Solutions for Class 12 Maths Chapter 9 – Differential Equations is based on the following topics:

  1. Formation of a Differential Equation whose General Solution is given.
  2. Procedure to form a differential equation that will represent a given family of curves.

Solving the third exercise will help the students improve their hold on these topics and also to score well when the questions are asked from it.

NCERT Solutions for Class 12 Maths Chapter 9 – Differential Equations Exercise 9.3

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Access Answers to NCERT Class 12 Maths Chapter 9 – Differential Equations Exercise 9.3 Page Number 391

In each of the Exercises 1 to 5, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

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Solution:-

NCERT Solutions for Class 12 Maths Chapter 9 - Image 31

2. y2 = a (b2 – x2)

Solution:-

NCERT Solutions for Class 12 Maths Chapter 9 - Image 32

NCERT Solutions for Class 12 Maths Chapter 9 - Image 33

3. y = ae3x + be-2x

Solution:-

NCERT Solutions for Class 12 Maths Chapter 9 - Image 34

NCERT Solutions for Class 12 Maths Chapter 9 - Image 35

4. y = e2x (a + bx)

Solution:-

From the question, it is given that y = e2x (a + b x)  … [we call it equation (i)]

Differentiating both sides with respect to x, we get

y’ = 2e2x(a + b x) + e2x × b … [equation (ii)]

Then, multiply equation (i) by 2 and afterwards, subtract it from equation (ii).

We have

y’ – 2y = e2x(2a + 2bx + b) – e2x (2a + 2bx)

y’ – 2y = 2ae2x + 2e2xbx + e2xb – 2ae2x – 2bxe2x

y’ – 2y = be2x … [equation (iii)]

Now, differentiating equation (iii) both sides,

We have

⇒ y’’ – 2y = 2be2x … [equation (iv)]

Then,

NCERT Solutions for Class 12 Maths Chapter 9 - Image 36

5. y = ex (a cos x + b sin x)

Solution:

From the question, it is given that y = ex(a cos x + b sin x)

… [we call it equation (i)]

Differentiating both sides with respect to x, we get

⇒y’ = ex(a cos x + b sin x) + ex(-a sin x + b cos x)

⇒ y’ = ex[(a + b)cos x – (a – b) sin x)] … [equation (ii)]

Now, differentiating equation (ii) both sides,

We have

y” = ex[(a + b) cos x – (a – b)sin x)] + ex[-(a + b)sin x – (a – b) cos x)]

On simplifying, we get

⇒ y” = ex[2bcosx – 2asinx]

⇒ y” = 2ex(b cos x – a sin x) … [equation (iii)]

Now, adding equations (i) and (iii), we get

NCERT Solutions for Class 12 Maths Chapter 9 - Image 37

6. Form the differential equation of the family of circles touching the y-axis at the origin.

Solution:

NCERT Solutions for Class 12 Maths Chapter 9 - Image 38

By looking at the figure, we can say that the centre of the circle touching the y-axis at the origin lies on the x-axis.

Let us assume (p, 0) to be the centre of the circle.

Hence, it touches the y-axis at the origin, and its radius is p.

Now, the equation of the circle with centre (p, 0) and radius (p) is

⇒ (x – p)2 + y2 = p2

⇒ x2 + p2 – 2xp + y2 = p2

Transposing p2 and – 2xp to RHS, it becomes – p2 and 2xp

⇒ x2 + y2 = p2 – p2 + 2px

⇒ x2 + y2 = 2px … [equation (i)]

Now, differentiating equation (i) both sides,

We have

⇒ 2x + 2yy’ = 2p

⇒ x + yy’ = p

Now, on substituting the value of ‘p’ in the equation, we get

⇒ x2 + y2 = 2(x + yy’)x

⇒ 2xyy’ + x2 = y2

Hence, 2xyy’ + x2 = y2 is the required differential equation.

7. Form the differential equation of the family of parabolas having a vertex at the origin and axis along the positive y-axis.

Solution:

The parabola having the vertex at the origin and the axis along the positive y-axis is

x2 = 4ay … [equation (i)]

NCERT Solutions for Class 12 Maths Chapter 9 - Image 39

NCERT Solutions for Class 12 Maths Chapter 9 - Image 40

8. Form the differential equation of the family of ellipses having foci on y-axis and centre at the origin.

Solution:

NCERT Solutions for Class 12 Maths Chapter 9 - Image 41

NCERT Solutions for Class 12 Maths Chapter 9 - Image 42

NCERT Solutions for Class 12 Maths Chapter 9 - Image 43

NCERT Solutions for Class 12 Maths Chapter 9 - Image 44

On simplifying,

⇒ -x (y’)2 – xyy” + yy’ = 0

⇒ xyy” + x (y’)2 – yy’ = 0

Hence, xyy” + x (y’)2 – yy’ = 0 is the required differential equation.

9. Form the differential equation of the family of hyperbolas having foci on the x-axis and centre at the origin.

Solution:

NCERT Solutions for Class 12 Maths Chapter 9 - Image 45

NCERT Solutions for Class 12 Maths Chapter 9 - Image 46

NCERT Solutions for Class 12 Maths Chapter 9 - Image 47

⇒ x (y’)2 + xyy” – yy’ = 0

⇒ xyy” + x(y’)2 – yy’ = 0

Hence, xyy” + x (y’)2 – yy’ = 0 is the required differential equation.

10. Form the differential equation of the family of circles having a centre on the y-axis and a radius 3 units.

Solution:

NCERT Solutions for Class 12 Maths Chapter 9 - Image 48

Let us assume the centre of the circle on the y-axis to be (0, a).

We know that the differential equation of the family of circles with centre at (0, a) and radius 3 is x2 + (y- a)2 = 32

⇒ x2 + (y- a)2 = 9 … [equation (i)]

Now, differentiating equation (i) both sides with respect to x,

⇒ 2x + 2(y – a) × y’ = 0 … [dividing both side by 2]

⇒ x + (y – a) × y’ = 0

Transposing x to the RHS, it becomes – x.

⇒ (y – a) × y’ = x

NCERT Solutions for Class 12 Maths Chapter 9 - Image 49

11. Which of the following differential equations has y = c1 ex + c2 e-x as the general solution?

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Solution:

NCERT Solutions for Class 12 Maths Chapter 9 - Image 51

Explanation:

NCERT Solutions for Class 12 Maths Chapter 9 - Image 52

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12. Which of the following differential equations has y = x as one of its particular solutions?

NCERT Solutions for Class 12 Maths Chapter 9 - Image 54

Solution:

NCERT Solutions for Class 12 Maths Chapter 9 - Image 55

Explanation:

NCERT Solutions for Class 12 Maths Chapter 9 - Image 56


Access Other Exercise Solutions of Class 12 Maths Chapter 9

Exercise 9.1 Solutions: 12 Questions

Exercise 9.2 Solutions: 12 Questions

Exercise 9.4 Solutions: 23 Questions

Exercise 9.5 Solutions: 17 Questions

Exercise 9.6 Solutions: 19 Questions

Miscellaneous Exercise on Chapter 9 Solutions: 18 Questions

Also, explore – 

NCERT Solutions for Class 12 Maths

NCERT Solutions for Class 12

NCERT Solutions

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