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# NCERT Solutions class 12 Maths Miscellaneous

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## 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

### 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 =

### 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),

### 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),

### 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

### 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),

### 12. Solve the differential equation:

Ans. Given: Differential equation

Comparing this equation with ,  P =  and Q =

I.F. =

The general solution is   (I.F.) =

### 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),

### 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

### 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

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.

### 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.

### 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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