## Chapter 22 – Differential Equations Exercise Ex. 22.1

Determine the order

and degree of the following differential equations. State also whether they

are linear or non-linear.

The order of a

differential equation is the order of the highest order derivative appearing

in the equation.

The degree of a

differential equation is the degree of the highest order derivative.

Consider the given

differential equation

In the above equation,

the order of the highest order derivative is 1.

So the differential

equation is of order 1.

In the above

differential equation, the power of the highest order derivative is 3.

Hence, it is a

differential equation of degree 3.

Since the degree of

the above differential equation is 3, more than one, it is a non-linear

differential equation.

## Chapter 22 – Differential Equations Exercise Ex. 22.10

dy = cos x (2 – y cosec x) dx

## Chapter 22 – Differential Equations Exercise Ex. 22.11

## Chapter 22 – Differential Equations Exercise Ex. 22.2

Form the differential equation having y = (sin^{-1}x)^{2} + A cos^{ -1} x + B, where A and B are arbitrary constants, as its general solution.

## Chapter 22 – Differential Equations Exercise Ex. 22.3

Show that y = e^{x}(A cos x + B sin x) is the solution of the differential equation

## Chapter 22 – Differential Equations Exercise Ex. 22.4

## Chapter 22 – Differential Equations Exercise Ex. 22.5

Solve the following differential equation:

(sin x + cos x)dy + (cos x – sin x) dx = 0

solve the following differential equation

## Chapter 22 – Differential Equations Exercise Ex. 22.6

Solve the following differential equation:

Solve the following differential equation:

## Chapter 22 – Differential Equations Exercise Ex. 22.7

Solve the following

differential equation:

ye^{x}^{/y} dx = (xe^{x}^{/y} + y^{2}) dy, y ¹ 0

(1 + y^{2}) tan^{-1} x dx + 2y (1 + x^{2})dy = 0

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

## Chapter 22 – Differential Equations Exercise Ex. 22.8

Solve the following differential equation.

## Chapter 22 – Differential Equations Exercise Ex. 22.9

Solve the following differential equation:

Solve the following differential equation:

## Chapter 22 – Differential Equations Exercise Ex. 22RE

## Chapter 22 – Differential Equations Exercise Ex. 22VSAQ

Write the differential equation

representing family of curves y = mx, where m is arbitrary constant.

## Chapter 22 – Differential Equations Exercise MCQ

Mark the correct alternative in each of the following

Correct option: (c)

- log y =kx
- y =kx
- xy =k
- y = k log x

Correct option: (b)

- Sin x
- Sec x
- Tan x
- Cos x

Correct option: (b)

- ½
- 2
- 3
- 4

Correct option:(b)

Degree is the power of highest order derivative.

Highest order is 2 and its power is 2.

Hence, degree of differential equation is 2.

- 4
- 2
- 5
- 10

NOTE: Answer not matching with back answer.

- x + y sin x =C
- x + y cos x = C
- y+ x ( sin x + cos x) = C
- y sin x = x + C

Correct option:(d)

The different equation obtained on eliminating A and B from y = A cos ωt + B sin ωt ,is

- Y
^{”}+ y^{‘}=0 - Y”- ω
^{2}y=0 - Y”= -ω
^{2}y - Y
^{”}+ y =0

Correct option: (c)

- x
^{2 }= y - y
^{2 }= x - x
^{2}= 2y - y
^{2 }= 2x

Correct option: (a)

The order of the different equation whose general solution is given by y =c_{1 }cos (2x +c_{2}) – (c_{3} +c_{4}) a^{x+c}_{5} + c_{6 } sin (x -c_{7}) is

- 3
- 4
- 5
- 2

Correct option: (c)

Here, constants are c_{1}, c_{2}, c_{3}, c_{4}, c_{5}, c_{6}.

But c_{3}+c_{4} is also constant. Hence, total 5 constants.

- a = b
- a = -b
- a =-2b
- a =2b

Correct option: (b)

Correct option: (a)

Correct option: (a)

- 1
- 2
- 3
- 4

Correct option: (a)

Differential equation contains only one constant hence,

Order of differential equation is 1.

The solution of the differential equation y_{1} y_{3}=y_{2}^{2} is

- x= C
_{1}e^{C}_{2}^{y}+C_{3} - y= C
_{1}e^{C}_{2}^{x}+C_{3} - 2x= C
_{1}e^{C}_{2}^{y}+C_{3} - None of these

Correct option: (b)

Correct option: (b)

Correct option: (d)

Correct option: (a)

- x( y + cos x) = sin x +C
- x( y – cos x) = sin x +C
- x( y + cos x) = cos x +C
- None of these

Correct option: (a)

The equation of the curve satisfying the differential equation y(x+y^{3}) dx = x(y^{3}-x) dy and passing through the point (1,1) is

- y
^{3}-2x+3x^{2}y =0 - y
^{3}+2x+3x^{2}y =0 - y
^{3}+2x-3x^{2}y =0 - None of these

Correct option: (c)

- Circles
- Straight lines
- Ellipses
- Parabolas

Correct option: (d)

Correct option: (b)

The different equation satisfied by ax^{2}+by^{2}=1 is

a. xyy_{2}+y_{1}^{2}+yy_{1}=0

b. xyy_{2}+xy_{1}^{2}-yy_{1}=0

c. xyy_{2}-xy_{1}^{2}+yy_{1}=0

d. none of these

Correct option: (b)

The different equation which represents the family of curves y = e^{Cx} is

- y
_{1}= C^{2}y - xy
_{1}– ln y =0 - x ln y = yy
_{1} - y ln y = xy
_{1}

Correct option: (d)

Note: log is considered same as ln.

- u = log x
- u = e
^{z} - u = (log z)
^{-1} - u = (log z)
^{2}

Correct option: (c)

Correct option:(a)

- m = 3, n = 3
- m = 3, n =2
- m = 3, n =5
- m =3, n =1

Correct option: (b)

Correct option: (d)

Correct option: (d)

The family of curves in which the subtangent at any point of a curve is double the abscisae, is given by

- x =Cy
^{2} - y =Cx
^{2} - x
^{2}=Cy^{2} - y =Cx

Correct option: (a)

The solution of the differential equation x dx +y dy =x^{2}y dy -y^{2} x dx , is

- x
^{2}-1 = C (1+y^{2}) - x
^{2}+1=C (1-y^{2}) - x
^{3}-1=C (1+y^{3}) - x
^{3}+1=C (1-y^{3})

Correct option:(a)

Correct option:

Correct option: (b)

- k =0
- k > 0
- k < 0
- none of these

Correct option:(c)

- tan
^{-1 }x-tan^{-1}y = tan^{-1}C - tan
^{-1 }y-tan^{-1}x = tan^{-1}C - tan
^{-1 }y ± tan^{-1}x = tan C - tan
^{-1 }y +tan^{-1}x = tan^{-1}C

Correct option: (d)

Correct option: (b)

Correct option:(d)

- p < q
- p = q
- p > q
- none of these

Correct option: (c)

Note: Answer not matching with back answer.

- x
- e
^{x} - log x
- log (log x)

Correct option: (c)

- sec x + tan x
- log (sec x+ tan x)
- e
^{sec}^{ x} - sec x

Correct option: (a)

(a) cos x

(b) tan x

(c) sec x

(d) sin x

Correct option: (c)

(a) 3

(b) 2

(c) 1

(d) Not defined

Correct option: (d)

Highest order derivative is 2 but equation cannot be expressed as a polynomial in differential equation.

Hence, it is not defined.

(a) 2

(b) 1

(c) 0

(d) Not defined

Correct option:(a)

Highest order of the derivative is 2.

The number of arbitrary constants in the general solution of differential equation of fourth order is

(a) 0

(b) 2

(c) 3

(d) 4

Correct option: (d)

In the general solution of differential equation of order n has n number of arbitrary constants.

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

(a) 3

(b) 2

(c) 1

(d) 0

Correct option: (d)

The number of arbitrary constants in the particular solution of a differential equation of third order is always zer0.

Correct option: (b)

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

Correct option: (c)

(a) e^{x} +e^{-y} =C

(b) e^{x }+ e^{y}= C

(c) e^{-x} +e^{y} =C

(d) e-^{x} +e^{-y} =C

Correct option: (a)

(a) y = vx

(b) v = yx

(c) x = vy

(d) x = v

Correct option: (c)

Which of the following is a homogeneous differential equation?

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

(b) xy dx -(x^{3}+y^{3})dy =0

(c) (x^{3}+2y^{2})dx+2xy dy =0

(d) y^{2 }dx+(x^{2}-xy-y^{2}) dy =0

Correct option: (d)

- e
^{-x} - e
^{-y} - 1/x
- x

Correct option: (c)

Correct option:(d)