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Install NowNCERT Solutions class 12 Maths Miscellaneous Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

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

## NCERT Solutions class 12 Maths Differential Equations

**1. For each of the differential equations given below, indicate its order and degree (if defined):**

**(i) **** **

**(ii) **** **

**(iii) **** **

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

The highest order derivative present in this differential equation is and hence order of this differential equation if 2.

The given differential equation is a polynomial equation in derivatives and highest power of the highest order derivative is 1.

Therefore, Order = 2, Degree = 1

**(ii)** Given: Differential equation

The highest order derivative present in this differential equation is and hence order of this differential equation if 1.

The given differential equation is a polynomial equation in derivatives and highest power of the highest order derivative is 3.

Therefore, Order = 1, Degree = 3

**(iii)** Given: Differential equation

The highest order derivative present in this differential equation is and hence order of this differential equation if 4.

The given differential equation is not a polynomial equation in derivatives therefore, degree of this differential equation is not defined.

Therefore, Order = 4, Degree not defined

**2. For each of the exercises given below verify that the given function (implicit or explicit) is a solution of the corresponding differential equation:**

**(i) **** **

**(ii) **** **

**(iii) **** **

**(iv) **** **

**Ans. (i)** The given function is ……….(i)

To verify: Function (i) is a solution of D.E. ……….(ii)

Differentiating both sides of eq. (i) w.r.t.

Again differentiating both sides w.r.t.

Putting from eq. (i), we have,

Therefore, Function given by eq. (i) is a solution of D.E. (ii).

**(ii)** The given function is ……….(i)

To verify: Function given by (i) is a solution of D.E. …….(ii)

From (i),

[By eq. (i)] ……….(iii)

[Using eq. (iii) and (i)]

Therefore, Function given by eq. (i) is a solution of D.E. (ii).

**(iii)** The given function is ………(i)

To verify: Function given by eq. (i) is a solution of D.E. …(ii)

From eq. (i),

[Using eq. (i)]

Therefore, Function given by eq. (i) is a solution of D.E. (ii).

**(iv)** The given function is ……….(i)

To verify: Function given by eq. (i) is a solution of D.E. ……(ii)

Differentiating both sides of eq. (i) w.r.t.

Putting from eq. (i), we get

Therefore, Function given by eq. (i) is a solution of D.E. (ii).

**3. Form the differential equation representing the family of curves **** where **** ia an arbitrary constant.**

**Ans. **Equation of the given family of curves is

……….(i)

Here number of arbitrary constants is one only

So, we will differentiate both sides of equation only once, w.r.t.

……….(ii)

Dividing eq. (i) by eq. (ii), we have

**4. Prove that **** is the general equation of the differential equation **** where **** is a parameter.**

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

Here each coefficient of and is of same degree, i.e., 3, therefore differential equation looks to be homogeneous.

……….(ii)

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

Putting

Putting these values in eq. (ii),

[Separating variables]

Integrating both sides, ……….(iii)

Now forming partial fraction of

=

=

= ……….(iv)

Comparing coefficients of like powers of

A – B – C = 1 ……….(v)

A + B – D = 0 ……….(vi)

A – B + C = –3 ……….(vii)

Constants A + B + D = 0 ………(viii)

Now eq. (v) – eq. (vii)

– 2C = 4 C = – 2

Eq. (vi) – eq. (viii)

– 2D = 0 D = 0

Putting C = – 2 in eq. (v), A – B + 2 = 1

A – B = –1 ……….(ix)

Putting D = 0 in eq. (vi) A + B = 0 ……….(x)

Adding eq. (ix) and (x) 2A = –1

A =

From eq. (x), B = –A =

Putting the values of A, B, C and D in eq. (iv), we have

=

Putting this value in eq. (iii),

Squaring both sides and cross-multiplying,

Putting

where

**5. For the differential equation of the family of the circles in the first quadrant which touch the coordinate axes.**

**Ans. **We know that the circle in the first quadrant which touches the co-ordinates axes has centre where is the radius of the circle.

Equation of the circle is

……….(i)

Differentiating with respect to

Substituting value of in eq. (i),

**6. Find the general solution of the differential equation **** **

**Ans. **Given: Differential Equation

Integrating both sides

### NCERT Solutions class 12 Maths Miscellaneous

**7. Show that the general solution of the differential equation **** is given by **** **

**Ans. **Given: Differential equation

Integrating both sides, ……….(i)

Now [Completing the squares]

Therefore,

=

Similarly,

Putting these values in eq. (i),

[Multiplying by ]

where

Multiplying every term in the numerator and denominator of L.H.S. by 3, and dividing every term by

where A =

### NCERT Solutions class 12 Maths Miscellaneous

**8. Find the equation of the curve passing through the point **** whose differential equation is **** **

**Ans. **Given: Differential equation

Integrating both sides,

……….(i)

Now, curve (i) passes through

Therefore, putting in eq. (i),

Putting in eq. (i),

### NCERT Solutions class 12 Maths Miscellaneous

**9. Find the particular solution of the differential equation **** given that **** when **** **

**Ans. **Given: Differential equation

Dividing every term by we have

Integrating both sides,

……….(i)

Now to evaluate , putting

=

Putting this value in eq. (i), ……….(ii)

Now putting in eq. (ii),

Putting in eq. (ii),

### NCERT Solutions class 12 Maths Miscellaneous

**10. Solve the differential equation: **** **

**Ans. **Given: Differential equation

……….(i)

It is not a homogeneous differential equation because of presence of only as a factor, yet it can be solved by putting i.e.,

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

### NCERT Solutions class 12 Maths Miscellaneous

**11. Find the particular solution of the differential equation **** given that **** when **

**Ans. **Given: Differential equation

……….(i)

Putting

Putting this value in eq. (i),

=

Integrating both sides,

Putting ,

……….(ii)

Now putting in eq. (ii),

Putting in eq. (ii),

### NCERT Solutions class 12 Maths Miscellaneous

**12. Solve the differential equation: **** **

**Ans. **Given: Differential equation

Comparing this equation with , P = and Q =

I.F. =

The general solution is (I.F.) =

### NCERT Solutions class 12 Maths Miscellaneous

**13. Find the particular solution of the differential equation **** given that **** when **** **

**Ans. **Given: Differential equation

Comparing this equation with , P = and Q =

I.F. =

The general solution is (I.F.) =

……….(i)

Now putting in eq. (i),

Putting in eq. (i),

### NCERT Solutions class 12 Maths Miscellaneous

**14. Find the particular solution of the differential equation **** given that **** when **

**Ans. **Given: Differential equation

Integrating both sides,

Putting

Putting ,

where C = ……….(i)

Putting in eq. (i), = C

C = 1

Putting C = 1 in eq. (i),

This solution may be written as

=

where expresses as an explicit function of

### NCERT Solutions class 12 Maths Miscellaneous

**15. The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20,000 in 1999 and 25,000 in the year 2004, what will be the population of the village in 2009?**

**Ans. **Let P be the population of the village at time

According to the question, Rate of increase of population of the village is proportional to the number of inhabitants.

where > 0 because of increase and is the constant of proportionality

[Separating variables]

Integrating both sides,

……….(i)

Now Population of the village was P = 20,000 in the year 1999.

Let us take the base year 1999 as

Putting and P = 20000 in eq. (i),

Now putting in eq. (i),

………..(ii)

Again Population of the village was P = 25,000 in the year 2004, when

Putting and P = 25000 in eq. (ii),

Putting value of in eq. (ii), ………..(iii)

To find the population in the year 2009,

Putting in eq. (iii),

25 x 1250 = 31250

**Choose the correct answer:**

**16. The general solution of the differential equation **** is:**

**(A) = C **

**(B) **

**(C) **

**(D) **

**Ans. **Given: Differential equation

[Separating variables]

Integrating both sides,

where C =

Therefore, option (C) is correct.

### NCERT Solutions class 12 Maths Miscellaneous

**17. The general equation of a differential equation of the type **** is:**

**(A) **** **

**(B) **

**(C) **

**(D) **

**Ans. **We know that general solution of differential equation of the type is

where =

Therefore, option (C) is correct.

### NCERT Solutions class 12 Maths Miscellaneous

**18. The general solution of the differential equation **** is:**

**(A) **

**(B) **

**(C) **

**(D) **

**Ans. **Given: Differential equation

Comparing with P = 1 and Q =

I.F. =

Solution is

Therefore, option (C) is correct.

## NCERT Solutions class 12 Maths Miscellaneous

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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