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NCERT Solutions class 12 Maths Exercise 9.3 (Ex 9.3) Chapter 9 Differential Equations

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

Free PDF download of NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.3 (Ex 9.3) and all chapter exercises at one place prepared by expert teacher as per NCERT (CBSE) books guidelines. Class 12 Maths Chapter 9 Differential Equations Exercise 9.3 Questions with Solutions to help you to revise complete Syllabus and Score More marks.

NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations (Ex 9.3) Exercise 9.3

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

1.

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

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

……….(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 a and b, therefore we will differentiate both sides two times w.r.t. x,

……….(ii)

Again differentiating w.r.t. x,

……….(iii)

Putting this value of -a from eq (i) in eq. (ii), we get

=>

3.

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

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

……….(ii)

Again differentiating w.r.t. x

=>d2ydx2=9ae3x+4be2x=>d2ydx2=9ae3x+4be−2x ……….(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 a and b, therefore we will differentiate both sides two times w.r.t. x

[By eq. (i)]……….(ii)

Again differentiating w.r.t. x,

=>   ……….(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 a and b, therefore we will differentiate both sides two times w.r.t. x

[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 y-axis at the origin must have its centre on x-axis, because x-axis being at right angles to y-axis is the normal or line of radius of the circle.

Therefore, the centre of the circle is (r,0) where r is the radius of the circle.

Equation of the required circle is

……….(i)

Here r is the only arbitrary constant.

differentiating w.r.t. x, we get

……….(ii)

Putting the value of 2r 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 y-axis.

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

……….(i)

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

……….(ii)

[From eq. (i)]

, which is the required differential equation.

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

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

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

……….(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 x-axis and centre at the origin.

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

……….(i)

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

……….(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 y-axis and radius 3 units.

Ans. We know that on y-axis,

Centre of the circle on y- axis is .

Equation of the circle having centre on y-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.