These Solutions are part of NCERT Solutions for Class 12 Maths . Here we have given NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations

Board |
CBSE |

Textbook |
NCERT |

Class |
Class 12 |

Subject |
Maths |

Chapter |
Chapter 9 |

Chapter Name |
Differential Equations |

Exercise |
Ex 9.1, Ex 9.2, Ex 9.3, Ex 9.4, Ex 9.5, Ex 9.6 |

Number of Questions Solved |
95 |

Category |
NCERT Solutions |

## NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations

### Chapter 9 Differential Equations Exercise 9.1

**Determine order and degree (if defined) of the differential equations given in Questions 1 to 10.**

**Question 1.**

**Solution:**

Order of the equation is 4

It is not a polynomial in derivatives so that it

has not degree.

**Question 2.**

**Solution:**

It is a D.E. of order one and degree one.

**Question 3.**

**Solution:**

Order of the equation is 2.

Degree of the equation is

**Question 4.**

**Solution:**

It is a D.E. of order 2 and degree undefined

**Question 5.**

**Solution:**

It is a D.E. of order 2 and degree 1.

**Question 6.**

**Solution:**

Order of the equation is 3

Degree of the equation is 2

**Question 7.**

**Solution:**

The highest order derivative is y.

Thus the order of the D.E. is 3.

The degree of D.E is 1

**Question 8.**

**Solution:**

The order of the D. E. = 1 (highest order derivative)

The degree of the D.E. = 1.

**Question 9.**

**Solution:**

The highest derivative is 2.

Order of the D.E. = 2.

Degree of the D. E = 1

**Question 10.**

**Solution:**

Order of the equation is 2

Degree of the equation is 1

**Question 11.**

The degree of the differential equation

(a) 3

(b) 2

(c) 1

(d) not defined

**Solution:**

The degree not defined.

Because the differential equation can not be written as a polynomial in all the differential coefficients.

Hence option (d) is correct.

**Question 12.**

The order of the differential equation

(a) 2

(b) 1

(c) 0

(d) not defined

**Solution:**

Thus order of the D.E. = 2

Hence option (a) is correct.

### Chapter 9 Differential Equations Exercise 9.2

**Verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation**

**Question 1.**

**Solution:**

**Question 2.**

**Solution:**

**Question 3.**

**Solution:**

**Question 4.**

**Solution:**

**Question 5.**

**Solution:**

**Question 6.**

**Solution:**

**Question 7.**

xy = logy + C,

**Solution:**

xy = logy + C,

**Question 8.**

**Solution:**

**Question 9.**

**Solution:**

**Question 10.**

**Solution:**

**Question 11.**

The number of arbitrary constants in the general solution of a differential equation of fourth order are:

(a) 0

(b) 2

(c) 3

(d) 4

**Solution:**

(b) The general solution of a differential equation of fourth order has 4 arbitrary constants.

Because it contains the same number of arbitrary constants as the order of differential equation.

**Question 12.**

The number of arbitrary constants in the particular solution of a differential equation of third order are:

(a) 3

(b) 2

(c) 1

(d) 0

**Solution:**

(d) Number of arbitrary constants = 0

Because particular solution is free from arbitrary constants.

### Chapter 9 Differential Equations Exercise 9.3

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

**Question 1.**

**Solution:**

Given that …(i)

differentiating (i) w.r.t x, we get

…(ii)

again differentiating w.r.t x, we get

which is the required differential equation

**Question 2.**

y² = a(b² – x²)

**Solution:**

given that

y² = a(b² – x²)…(i)

**Question 3.**

y = ae^{3x}+be^{-2x}

**Solution:**

Given that

y = ae^{3x}+be^{-2x} …(i)

**Question 4.**

y = e^{2x} (a+bx)

**Solution:**

y = e^{2x} (a+bx)

**Question 5.**

y = e^{x}(a cosx+b sinx)

**Solution:**

The curve y = e^{x}(a cosx+b sinx) …(i)

differentiating w.r.t x

**Question 6.**

Form the differential equation of the family of circles touching the y axis at origin

**Solution:**

The equation of the circle with centre (a, 0) and radius a, which touches y- axis at origin

**Question 7.**

Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.

**Solution:**

The equation of parabola having vertex at the origin and axis along positive y-axis is

**Question 8.**

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

**Solution:**

The equation of family ellipses having foci at y- axis is

**Question 9.**

Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.

**Solution:**

Equation of the hyperbola is

Differentiating both sides w.r.t x

which is the req. differential eq. of the hyperbola.

**Question 10.**

Form the differential equation of the family of circles having centre on y-axis and radius 3 units

**Solution:**

Let centre be (0, a) and r = 3

Equation of circle is

x² + (y – a)² = 9 …(i)

Differentiating both sides, we get

which is required equation

**Question 11.**

Which of the following differential equation has as the general solution ?

(a)

(b)

(c)

(d)

**Solution:**

(b)

**Question 12.**

Which of the following differential equations has y = x as one of its particular solution ?

(a)

(b)

(c)

(d)

**Solution:**

(c) y = x

### Chapter 9 Differential Equations Exercise 9.4

**For each of the following D.E in Q. 1 to 10 find the general solution:**

**Question 1.**

**Solution:**

integrating both sides, we get

**Question 2.**

**Solution:**

**Question 3.**

**Solution:**

which is required solution

**Question 4.**

sec² x tany dx+sec² y tanx dy = 0

**Solution:**

we have

sec² x tany dx+sec² y tanx dy = 0

**Question 5.**

**Solution:**

we have

Integrating on both sides

**Question 6.**

**Solution:**

integrating on both side we get

which is required solution

**Question 7.**

y logy dx – x dy = 0

**Solution:**

integrating we get

**Question 8.**

**Solution:**

**Question 9.**

solve the following

**Solution:**

integrating both sides we get

**Question 10.**

**Solution:**

we can write in another form

**Find a particular solution satisfying the given condition for the following differential equation in Q.11 to 14.**

**Question 11.**

**Solution:**

here

integrating we get

**Question 12.**

**Solution:**

**Question 13.**

**Solution:**

**Question 14.**

**Solution:**

=> logy = logsecx + C

When x = 0, y = 1

=> log1 = log sec0 + C => 0 = log1 + C

=> C = 0

∴ logy = log sec x

=> y = sec x.

**Question 15.**

Find the equation of the curve passing through the point (0,0) and whose differential equation

**Solution:**

**Question 16.**

For the differential equation find the solution curve passing through the point (1,-1)

**Solution:**

The differential equation is

or xydy=(x + 2)(y+2)dx

**Question 17.**

Find the equation of a curve passing through the point (0, -2) given that at any point (pc, y) on the curve the product of the slope of its tangent and y-coordinate of the point is equal to the x-coordinate of the point

**Solution:**

According to the question

0, – 2) lies on it.c = 2

∴ Equation of the curve is : x² – y² + 4 = 0.

**Question 18.**

At any point (x, y) of a curve the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (-4,-3) find the equation of the curve given that it passes through (- 2,1).

**Solution:**

Slope of the tangent to the curve =

slope of the line joining (x, y) and (- 4, – 3)

**Question 19.**

The volume of a spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and offer 3 seconds it is 6 units. Find the radius of balloon after t seconds.

**Solution:**

Let v be volume of the balloon.

**Question 20.**

In a bank principal increases at the rate of r% per year. Find the value of r if Rs 100 double itself in 10 years

**Solution:**

Let P be the principal at any time t.

According to the problem

**Question 21.**

In a bank principal increases at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years

**Solution:**

Let p be the principal Rate of interest is 5%

**Question 22.**

In a culture the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000 if the rate of growth of bacteria is proportional to the number present

**Solution:**

Let y denote the number of bacteria at any instant t • then according to the question

**Question 23.**

The general solution of a differential equation is

(a)

(b)

(c)

(d)

**Solution:**

(a)

### Chapter 9 Differential Equations Exercise 9.5

**Show that the given differential equation is homogeneous and solve each of them in Questions 1 to 10**

**Question 1.**

(x²+xy)dy = (x²+y²)dx

**Solution:**

(x²+xy)dy = (x²+y²)dx

**Question 2.**

**Solution:**

**Question 3.**

(x-y)dy-(x+y)dx=0

**Solution:**

**Question 4.**

(x²-y²)dx+2xy dy=0

**Solution:**

**Question 5.**

**Solution:**

**Question 6.**

**Solution:**

**Question 7.**

**Solution:**

**Question 8.**

**Solution:**

**Question 9.**

**Solution:**

**Question 10.**

**Solution:**

**For each of the following differential equation in Q 11 to 15 find the particular solution satisfying the given condition:**

**Question 11.**

(x + y) dy+(x – y)dx = 0,y = 1 when x = 1

**Solution:**

given

(x + y) dy+(x – y)dx = 0

**Question 12.**

x²dy+(xy+y²)dx=0, y=1 when x=1

**Solution:**

f(x,y) is homogeneous

∴ put y = vx

**Question 13.**

**Solution:**

**Question 14.**

**Solution:**

which is a homogeneous differential equation

**Question 15.**

**Solution:**

…(i)

**Question 16.**

A homogeneous equation of the form can be solved by making the substitution,

(a) y=vx

(b) v=yx

(c) x=vy

(d) x=v

**Solution:**

(c) option x = vy

**Question 17.**

Which of the following is a homogeneous differential equation?

(a) (a) (4x + 6y + 5)dy-(3y + 2x + 4)dx = 0

(b)

(c)

(d)

**Solution:**

(d)

### Chapter 9 Differential Equations Exercise 9.6

**Find the general solution of the following differential equations in Q.1 to 12**

**Question 1.**

**Solution:**

Given equation is a linear differential equation of the form ;

Here, P = 2, Q = sin x

**Question 2.**

**Solution:**

Here P = 3,

which is required equation

**Question 3.**

**Solution:**

**Question 4.**

**Solution:**

Here, P = secx, Q = tanx;

= sec x + tan x

i.e., The solu. is y.× I.F. = ∫Q × I.F. dx + c

or y × (secx+tanx) = ∫tanx(secx+tanx)dx+c

Reqd. sol. is

∴ y(secx + tanx) = (secx + tanx)-x + c

**Question 5.**

**Solution:**

⇒ integrating factor =

**Question 6.**

**Solution:**

Here P = and Q = x logx

**Question 7.**

**Solution:**

**Question 8.**

(1+x²)dy+2xy dx = cotx dx(x≠0)

**Solution:**

(1+x²)dy+2xy dx = cotx dx

**Question 9.**

**Solution:**

**Question 10.**

**Solution:**

**Question 11.**

**Solution:**

**Question 12.**

**Solution:**

**For each of the following Questions 13 to is find a particular solution, satisfying the given condition:**

**Question 13.**

**Solution:**

**Question 14.**

**Solution:**

**Question 15.**

**Solution:**

Here P = -3cot x

Q = sin 2x

**Question 16.**

Find the equation of the curve passing through the origin given that the slope of the tangent to the curve at any point (x,y) is equal to the sum of the coordinates of the point

**Solution:**

**Question 17.**

Find the equation of the curve passing through the point (0, 2) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5

**Solution:**

By the given condition

**Question 18.**

The integrating factor of the differential equation

(a)

(b)

(c)

(d) x

**Solution:**

(c)

**Question 19.**

The integrating factor of the differential equation (-1<y<1) is

(a)

(b)

(c)

(d)

**Solution:**

(d)

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