# NCERT Solutions class 12 Maths Exercise 9.3

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## NCERT Solutions class 12 Maths Differential Equations

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

1.

Ans. Given: Equation of the family of curves   ……….(i)

Here there are two arbitrary constants  and  therefore we will differentiate both sides two times w.r.t.

……….(ii)

Again differentiating w.r.t. ,

Multiplying both sides by   , which the required differential equation.

2.

Ans. Given: Equation of the family of curves  ……….(i)

Here there are two arbitrary constants  and  therefore we will differentiate both sides two times w.r.t.

……….(ii)

Again differentiating w.r.t. ,

……….(iii)

Putting this value of  in eq. (ii), we get

3.

Ans. Given: Equation of the family of curves  ……….(i)

Here there are two arbitrary constants  and  therefore we will differentiate both sides two times w.r.t.

……….(ii)

Again differentiating w.r.t. ,  ……….(iii)

Multiplying eq. (i) by 3 and subtracting eq. (ii) from it, we get

……….(iv)

Again multiplying eq. (ii) by 3 and subtracting it from eq. (iii), we get

……….(v)

Now, eq. (v) + 2 eq. (iv) gives,

, which is required differential equation.

4.

Ans. Given: Equation of the family of curves  ……….(i)

Here there are two arbitrary constants  and  therefore we will differentiate both sides two times w.r.t.

[By eq. (i)]……….(ii)

Again differentiating w.r.t. ,  ……….(iii)

Now from eq. (ii),

Putting this value of  in eq. (iii),

5.

Ans. Given: Equation of the family of curves  ……….(i)

Here there are two arbitrary constants  and  therefore we will differentiate both sides two times w.r.t.

[By eq. (i)]……….(ii)

Again differentiating w.r.t. ,

[By eq. (i)]

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

Ans. It is clear that if a circle touches axis at the origin must have its centre on axis, because axis being at right angles to axis is the normal or line of radius of the circle.

Therefore, the centre of the circle is  where  is the radius of the circle.

Equation of the required circle is

……….(i)

Here  is the only arbitrary constant.

differentiating w.r.t. , we get

……….(ii)

Putting the value of  from eq. (ii) in eq. (i), we get

, which is the required differential equation.

### 7. Find the differential equation of the family of parabolas having vertex at origin and axis along positive axis.

Ans. We know that equation of parabolas having vertex at origin and axis along positive axis is   ……….(i)

Here  is the only arbitrary constant. Therefore differentiating w.r.t. , we get

……….(ii)

[From eq. (i)]

, which is the required differential equation.

### 8. Form the differential equation of family of ellipse having foci on axis and centre at the origin.

Ans. We know that equation of ellipse having foci on axis i.e., vertical ellipse with major axis as axis is   ………..(i)

Here there are two arbitrary constants  and  therefore we will differentiate both sides two times w.r.t.

……….(ii)

Again differentiating w.r.t. ,   ……….(iii)

Putting the value of  from eq. (iii), in eq. (ii), we get

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

Ans. We know that equation of hyperbolas having foci on axis and centre at origin is

……….(i)

#### Here there are two arbitrary constants  and  therefore we will differentiate both sides two times w.r.t.

……….(ii)

Again differentiating w.r.t. ,

……….(iii)

Dividing eq. (iii) by eq. (ii), we get

, which is required differential equation.

### 10. Form the differential equation of the family of circles having centres on axis and radius 3 units.

Ans. We know that on axis,

Centre of the circle on axis is .

Equation of the circle having centre on axis an radius  unit is

……….(i)

Here  is the only arbitrary constant, therefore we will differentiate only once.

Putting this value of  in eq. (i), we get

### 11. Which of the following differential equation has  as the general solution:

(A)

(B)

(C)

(D)

Ans. Given:  ……….(i)

[From eq. (i)]

Therefore, option (B) is correct.

### 12. Which of the following differential equations has  as one of its particular solutions:

(A)

(B)

(C)

(D)

Ans. Given:

On putting these values in the given option, we get the correct answer in option (C).

L.H.S. of differential equation of option (C) =

=

=

= R.H.S. of option (C)

Therefore, option (C) is correct.

## NCERT Solutions class 12 Maths Exercise 9.3

NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 20 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide

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