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## NCERT Solutions class 12 Maths Differential Equations

**In each of the following Questions 1 to 5, show that the differential equation is homogenous and solve each of them:**

**1. **** **

**Ans. **Given: Differential equation ……….(i)

Here degree of each coefficients of and is same therefore, it is homogenous.

…..(ii)

F , therefore the given differential equation is homogeneous.

Putting

Putting value of and in eq. (ii),

[Separating variables]

Integrating both sides,

Putting

where C =

**2. **** **

**Ans. **Given: Differential equation

……….(i)

Therefore, eq. (i) is homogeneous.

Putting

Putting value of and in eq. (i)

[Separating variables]

Integrating both sides,

Putting ,

**3. **** **

**Ans. **Given: Differential equation ……….(i)

This given equation is homogeneous because each coefficients of and is of degree 1.

Putting

….(ii)

Putting value of and in eq. (ii)

[Separating variables]

Integrating both sides,

Putting ,

**4. **** **

**Ans. **Given: Differential equation –

This equation is homogeneous because degree of each coefficient of and is same i.e., 2

……….(ii)

Therefore, the given equation is homogeneous.

Put

Putting these values of and in eq. (ii), we get

=

Integrating both sides,

Put ,

**5. **** **

**Ans. **Given: Differential equation

= ……….(i)

Therefore, the given differential equation is homogeneous as all terms of and are of same degree i.e., degree 2.

Putting

Putting these values of and in eq. (i), we get

[Separating variables]

Integrating both sides,

=

Putting , =

Multiplying within logs by in L.H.S., =

**NCERT Solutions class 12 Maths Exercise 9.5**

**In each of the Questions 6 to 10, show that the given differential equation is homogeneous and solve each of them:**

**6. **** **

**Ans. **Given: Differential equation

[Dividing by ]

Therefore given differential equation is homogeneous.

Putting

Putting these values of and in eq. (i), we get

Integrating both sides,

Putting ,

**7. **** **

**Ans. **Given: Differential equation

……….(i)

Therefore, the given differential equation is homogeneous.

Putting

Putting these values of and in eq. (i), we get

[Separating variables]

Integrating both sides,

=

Putting where C =

**8. **** **

**Ans. **Given: Differential equation

= ……….(i)

Therefore, the given differential equation is homogeneous.

Putting

Putting these values of and in eq. (i), we get

Integrating both sides,

[putting ]

where

**9. **** **

**Ans. **Given: Differential equation

……….(i)

Therefore, the given differential equation is homogeneous.

Putting

#### Putting these values of and in eq. (i), we get

=

#### Integrating both sides,

where C =

[Putting ]

**10. **** **

**Ans. **Given: Differential equation

[Dividing by ]

……….(i)

Therefore, it is a homogeneous.

Now putting

#### Putting these values of and in eq. (i), we have

[Separating variables]

Integrating both sides,

Now putting ,

C where C =

**NCERT Solutions class 12 Maths Exercise 9.5**

**For each of the differential equations in Questions from 11 to 15, find the particular solution satisfying the given condition**

**11. **** when **** **

**Ans. **Given: Differential equation when …..(i)

……….(ii)

#### Therefore the given differential equation is homogeneous because each coefficient of and is same i.e., degree 2.

Putting

Putting these values of and in eq. (ii), we have

[Separating variables]

Integrating both sides,

Now putting

……….(iii)

Now again given when , therefore putting these values in eq. (iii),

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

**12. **** when **

**Ans. **Given: Differential equation

……….(i)

#### Therefore the given differential equation is homogeneous.

Putting

Putting these values of and in eq. (i), we have

Integrating both sides,

Putting

where C = ……….(ii)

Now putting and in eq. (ii), we get 1 = 3C

Putting value of C in eq. (ii),

**13. **** when **

**Ans. **Given: Differential equation

= ……….(i)

Therefore, the given differential equation is homogeneous.

Putting

Putting these values of and in eq. (i), we have

[Separating variables]

Integrating both sides,

[Putting ] ……….(ii)

Now putting in eq. (ii),

Putting the value of in eq. (ii),

**14. **** when **

**Ans. **Given: Differential equation

………(i)

Therefore, the given differential equation is homogeneous.

Putting

Putting these values of and in eq. (i), we have

[Separating variables]

Integrating both sides,

[Putting ] ……….(ii)

Now putting in eq. (ii),

Putting the value of in eq. (ii),

**15. **** when **

**Ans. **Given: Differential equation ……….(i)

……….(ii)

Therefore the given differential equation is homogeneous because each coefficient of and is same i.e., degree 2.

Putting

#### Putting these values of and in eq. (ii), we have

[Separating variables]

Integrating both sides,

[Putting ]

Now putting in ,

Again putting , in , we get

**NCERT Solutions class 12 Maths Exercise 9.5**

**Choose the correct answer:**

**16. A homogeneous differential equation of the form **** can be solved by making the substitution:**

**(A) **

**(B) **

**(C) **

**(D) **

**Ans. **We know that a homogeneous differential equation of the form can be solved by the substitution i.e.,

Therefore, option (C) is correct.

**NCERT Solutions class 12 Maths Exercise 9.5**

**17. Which of the following is a homogeneous differential equation:**

**(A) **** **

**(B) **** **

**(C) **** **

**(D) **** **

**Ans. **Out of the given four options, option (D) is the only option in which all coefficients of and are of same degree i.e., 2. It may be noted that is a term of second degree.

Hence differential equation in option (D) is Homogeneous differential equation.

## NCERT Solutions class 12 Maths Exercise 9.5

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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In Q3 it should be 1+v² not 1-v²