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**Download NCERT solutions for Differential Equations as PDF**

## 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)

[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)

[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)

……….(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)

……….(ii)

Again differentiating w.r.t. ,

……….(iii)

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

, which is required differential equation.

### NCERT Solutions class 12 Maths Exercise 9.3

**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

### NCERT Solutions class 12 Maths Exercise 9.3

**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.

### NCERT Solutions class 12 Maths Exercise 9.3

**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

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