##### Question 1:

Determine order and degree (if defined) of differential equations given in

##### Answer:

Order Degree

4 Not a polynomial in derivatives

##### Question 2:

Determine order and degree (if defined) of differential equations given in y' + 5y = 0.

##### Answer:

Order Degree

1 1

##### Question 3:

Determine order and degree (if defined) of differential equations given in

##### Answer:

Order Degree

2 1

##### Question 4:

Determine order and degree (if defined) of differential equations given in

##### Answer:

Order Degree

2 Not a polynomial in derivatives

##### Question 5:

Determine order and degree (if defined) of differential equations given in

##### Answer:

Order Degree

2 1

##### Question 6:

Determine order and degree (if defined) of differential equations given in

(y''')^{2} + (y'')^{3} + (y')^{4} + y^{5} = 0.

##### Answer:

Order Degree

3 2

##### Question 7:

Determine order and degree (if defined) of differential equations given in

y''' + 2y'' + y' = 0.

##### Answer:

Order Degree

3 1

##### Question 8:

Determine order and degree (if defined) of differential equations given in

y'+ y = 0.

##### Answer:

Order Degree

1 1

##### Question 9:

Determine order and degree (if defined) of differential equations given in

y''+ (y')^{2} + 2y = 0.

##### Answer:

Order Degree

2 1

##### Question 10:

Determine order and degree (if defined) of differential equations given in

y''+ 2y'+ sin y = 0.

##### Answer:

Order Degree

2 1

##### Question 11:

The degree of the differential equation :

- 3
- 2
- 1
- not defined

##### Answer:

not defined

##### Question 12:

The order of the differential equation :

- 2
- 1
- 0
- Not defined.

##### Answer:

2

##### Question 13:

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

y = e^{x} + 1 : y''– y'= 0

##### Answer:

We have : y = ex + 1.

$\therefore $ y' = e^{x} and y'' = e^{x} ... (1)

Now y''– y' = e^{x} – e^{x} [Using (1)]

= 0, which is true.

##### Question 14:

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

y = x^{2} + 2x + C : y'– 2x – 2 = 0.

##### Answer:

We have : y = x^{2} + 2x + C.

$\therefore $ y'= 2x + 2

$\Rightarrow $ y'– 2x – 2 = 0, which is true.

##### Question 15:

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

y = cos x + C : y' sin x = 0

##### Answer:

We have : y = cos x + C.

$\therefore $ y' = – sin x + C

$\Rightarrow $ y' + sin x = 0, which is true.

##### Question 16:

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

##### Answer:

##### Question 17:

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

y = Ax : xy'= y (x $\ne $ 0)

##### Answer:

We have : y= Ax ...(1)

$\therefore $ y'= A.

Putting the value of A in (1),

y = y' x $\Rightarrow $ xy' = y,

which is true.

##### Question 18:

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

##### Answer:

We have : y = x sin x ...(1)

$\therefore $ y' = x cos x + sin x . 1

$\Rightarrow $ y'= sin x + x cos x ...(2)

##### Question 19:

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

##### Answer:

##### Question 20:

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

y – cos x = x : (y sin y + cos y + x)y'= y.

##### Answer:

We have : y – cos y = x ...(1)

Diff. w.r.t. x, y'+ sin y . y'= 1

$\Rightarrow $ (1 + sin y) y'= 1

$\Rightarrow $ (y + y sin y)y' = y ...(2) [Multiplying by y]

From (1), y = x + cos y.

Putting in (2), (x + cos y + y sin y) y'= y

$\Rightarrow $(y sin y + cos y + x) y'= y, which is true.

##### Question 21:

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

x + y = tan^{–1} y : y^{2}y'+ y^{2} + 1 = 0

##### Answer:

We have : x + y = tan^{–1} y.

^{2}) (1 + y') = y'

$\Rightarrow $ 1 + y

^{2}+ y'(1 + y

^{2}) = y'

$\Rightarrow $ 1 + y

^{2}+ y'(1 + y

^{2}– 1) = 0

$\Rightarrow $ 1 + y

^{2}+ y'y2 = 0

$\Rightarrow $ y

^{2}y' + y

^{2}+ 1 = 0, which is true.

##### Question 22:

##### Answer:

##### Question 23:

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

- 0
- 2
- 3
- 4

##### Answer:

4

##### Question 24:

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

- 3
- 2
- 1
- 0

##### Answer:

0

##### Question 25:

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b :

##### Answer:

##### Question 26:

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b :

y^{2} = a (b^{2} – x^{2})

##### Answer:

We have : y^{2} = a (b^{2} – x^{2}) ...(1)

Diff. w.r.t. x, 2yy' = a (0 – 2x)

$\Rightarrow $ yy' = – ax ...(2)

Again diff. w.r.t. x, yy'+ y'^{2} = – a ...(3)

Dividing (3) by (2),

^{2}) = yy'

$\Rightarrow $ xyy''+ x y'

^{2}– yy' = 0,

##### Question 27:

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b :

y = ae^{3x} + be^{–2x}

##### Answer:

We have : y = ae^{3x} + be^{– 2x} ...(1)

Diff. w.r.t. x, y' = 3ae^{3x} – 2b e^{–2x} ...(2)

Again diff. w.r.t. x, y' = 9ae^{3x} + 4be– 2x ...(3)

Multiplying (1) by 2,

2y = 2ae^{3x} + 2be^{–2x} ...(4)

Adding (2) and (4), y''+ 2y = 5ae^{3x}

##### Question 28:

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b :

y = e2x (a + bx)

##### Answer:

We have : y = e^{2x} (a + bx) ...(1)

Diff. w.r.t. x, y'= e^{2x} (b) + 2e^{2x} (a + bx)

$\Rightarrow $ y'= e^{2x} (2a + b + 2bx) ...(2)

Multiplying (1) by 2, 2y = e^{2x} (2a + 2bx) ...(3)

Subtracting (3) from (2),

y'– 2y = be^{2x} ...(4)

Again diff. w.r.t x, y''– 2y'= 2be^{2x} ...(5)

##### Question 29:

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b :

y = e^{x} (a cos x + b sin x).

##### Answer:

We have : y = e^{x} (a cos x + b sin x) ...(1)

Diff. w.r.t. x,

y''= e^{x} (– a sin x + b cos x)

+ e^{x} (a cos x + b sin x)

y''= e^{x} [(a + b) cos x – (a – b) sin x] ...(2)

Again diff. w.r.t. x,

y''= e^{x} [– (a + b) sin x – (a – b) cos x]

+ ex [(a + b) cos x – (a – b) sin x]

$\Rightarrow $ y'' = e^{x} [2b cos x – 2 a sin x]

##### Question 30:

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

##### Answer:

Let ($\mathrm{\alpha}$, 0) be the centre of any member of the circles. Then the equation of the family of circles is :

(x – $\mathrm{\alpha}$)^{2} + y^{2} =$\mathrm{\alpha}$^{2} $\Rightarrow $x^{2} + y^{2} – 2x = 0 ...(1)

##### Question 31:

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

##### Answer:

Let the equation of parabolas be :

x^{2} = 4ay ...(1)

Diff. w.r.t. x,

2x = 4ay'

##### Question 32:

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

##### Answer:

##### Question 33:

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

##### Answer:

##### Question 34:

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

##### Answer:

Let the equation of the family of circles be :

x^{2} + (y – $\mathrm{\alpha}$)^{2} = 9 ...(1)

##### Question 35:

Which of the following differential equation has y = C_{1} e^{x} + C_{2} e^{–x} as the general solution ?

##### Answer:

##### Question 36:

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

##### Answer:

##### Question 37:

For each of the differential equations, find the general solution :

##### Answer:

##### Question 38:

For each of the differential equations, find the general solution :

##### Answer:

##### Question 39:

For each of the differential equations, find the general solution :

##### Answer:

##### Question 40:

For each of the differential equations, find the general solution :

sec^{2} x tan y dx + sec^{2} y tan x dy = 0.

##### Answer:

$\Rightarrow $ log | tan x tan y | = log | C |$\Rightarrow $ tan x tan y = C,

which is the reqd. solution.

##### Question 41:

For each of the differential equations, find the general solution :

(e^{x} + e^{– x}) dy – (e^{x} – e^{– x}) dx = 0.

##### Answer:

We have : (e^{x} + e^{–x}) dy – (e^{x} – e^{–x}) dx = 0

##### Question 42:

For each of the differential equations, find the general solution :

##### Answer:

##### Question 43:

For each of the differential equations, find the general solution :

y log y dx – x dy = 0.

##### Answer:

##### Question 44:

For each of the differential equations, find the general solution :

##### Answer:

##### Question 45:

For each of the differential equations, find the general solution :

##### Answer:

##### Question 46:

For each of the differential equations, find the general solution :

e^{x} tan y dx + (1 – e^{x}) sec^{2} y dy = 0.

##### Answer:

##### Question 47:

For each of the differential equations find a particular solution satisfying the given condition :

##### Answer:

##### Question 48:

For each of the differential equations find a particular solution satisfying the given condition :

##### Answer:

##### Question 49:

For each of the differential equations find a particular solution satisfying the given condition :

##### Answer:

##### Question 50:

For each of the differential equations find a particular solution satisfying the given condition :

##### Answer:

##### Question 51:

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

y' = e^{x} sin x.

##### Answer:

##### Question 52:

##### Answer:

##### Question 53:

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

##### Answer:

##### Question 54:

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

##### Answer:

By the question,

##### Question 55:

The volume of a spherical balloon is being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units, find the radius of the balloon after ‘t’ seconds.

##### Answer:

Let ‘r’ be the radius of spherical balloon after time ‘t’.

##### Question 56:

In a bank, principal increases at the rate of r% per year. Find the value of ‘r’ if ₹ 100 double itself in10 years (log_{e} 2 = 0·6931).

##### Answer:

Let ‘P’ be the principal at any time t.

##### Question 57:

In a bank, principal, increases continuously at the rate of 5% per year. An amount of ₹ 1000 is deposited with this bank, how much will it worth after 10 years. (e^{0·5} = 1·648)

##### Answer:

Let ‘P’ be the principal at any time t.

##### Question 58:

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 culture reach 2,00,000, if the rate of growth of bacteria is proportional to the number present ?

##### Answer:

##### Question 59:

- e
^{x}+ e^{–y}= C - e
^{x}+ e^{y}= C - e
^{–x}+ e^{y}= C - e
^{–x}+ e^{–y}= C

##### Answer:

e^{x} + e^{–y} = C

##### Question 60:

Show that the given differential equation is homogeneous and solve each of them :

(x^{2} + xy) dy = (x^{2} + y^{2}) dx.

##### Answer:

##### Question 61:

Show that the given differential equation is homogeneous and solve each of them :

##### Answer:

##### Question 62:

Show that the given differential equation is homogeneous and solve each of them :

(x – y) dy – (x + y) dx = 0

##### Answer:

##### Question 63:

Show that the given differential equation is homogeneous and solve each of them :

(x^{2} – y^{2}) dx + 2xy dy = 0.

##### Answer:

##### Question 64:

Show that the given differential equation is homogeneous and solve each of them :

##### Answer:

##### Question 65:

Show that the given differential equation is homogeneous and solve each of them :

##### Answer:

##### Question 66:

Show that the given differential equation is homogeneous and solve each of them :

##### Answer:

##### Question 67:

Show that the given differential equation is homogeneous and solve each of them :

##### Answer:

##### Question 68:

Show that the given differential equation is homogeneous and solve each of them :

##### Answer:

##### Question 69:

Show that the given differential equation is homogeneous and solve each of them :

##### Answer:

##### Question 70:

For each of the differential equations find the particular solution satisfying the given condition :

(x + y) dy + (x – y) dx = 0 ;

y = 1 when x = 1.

##### Answer:

##### Question 71:

For each of the differential equations find the particular solution satisfying the given condition :

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

y = 1 when x = 1.

##### Answer:

##### Question 72:

For each of the differential equations find the particular solution satisfying the given condition :

##### Answer:

##### Question 73:

For each of the differential equations find the particular solution satisfying the given condition :

##### Answer:

##### Question 74:

For each of the differential equations find the particular solution satisfying the given condition :

##### Answer:

##### Question 75:

- y = vx
- y = yx
- x = vy
- x = y

##### Answer:

x = vy

##### Question 76:

Which of the following is a homogeneous differential equation ?

- (4x + 6y + 5) dy – (3xy + 2x + 4) dx
- (xy) dx – (x
^{3}+ y^{3}) dy = 0 - (x
^{3}+ 2y^{2}) dx + 2xy dy = 0 - y2dx + (x
^{2}– xy – y^{2}) dy = 0.

##### Answer:

y^{2}dx + (x^{2} – xy – y^{2}) dy = 0.

##### Question 77:

For each of the differential equations find the general solutions :

##### Answer:

##### Question 78:

For each of the differential equations find the general solutions :

##### Answer:

##### Question 79:

For each of the differential equations find the general solutions :

##### Answer:

##### Question 80:

For each of the differential equations find the general solutions :

##### Answer:

##### Question 81:

For each of the differential equations find the general solutions :

##### Answer:

##### Question 82:

For each of the differential equations find the general solutions :

##### Answer:

##### Question 83:

For each of the differential equations find the general solutions :

##### Answer:

##### Question 84:

For each of the differential equations find the general solutions :

(1 + x^{2}) dy + 2xy dx = cot x dx (x $\ne $ 0).

##### Answer:

##### Question 85:

For each of the differential equations find the general solutions :

##### Answer:

##### Question 86:

For each of the differential equations find the general solutions :

##### Answer:

##### Question 87:

For each of the differential equations find the general solutions :

y dx + (x – y^{2}) dy = 0.

##### Answer:

##### Question 88:

For each of the differential equations find the general solutions :

##### Answer:

##### Question 89:

For each of the differential equations find a particular solution satisfying the given conditions :

##### Answer:

##### Question 90:

For each of the differential equations find a particular solution satisfying the given conditions :

##### Answer:

##### Question 91:

For each of the differential equations find a particular solution satisfying the given conditions :

##### Answer:

##### Question 92:

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

##### Answer:

##### Question 93:

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

##### Answer:

By the question,

##### Question 94:

- e
^{–x} - e
^{–y} - x

##### Answer:

##### Question 95:

The Integrating Factor of the differential equation :

##### Answer:

##### Question 96:

The degree of the differential equation :

- 3
- 2
- 1
- not defined

##### Answer:

not defined

##### Question 97:

The order of the differential equation :

- 2
- 1
- 0
- not defined

##### Answer:

2

##### Question 98:

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

- 0
- 2
- 3
- 4

##### Answer:

4

##### Question 99:

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

- 3
- 2
- 1
- 0

##### Answer:

0

##### Question 100:

Which of the following differential equations has y = c^{1 ex + c2 e–x as the general solution ?}

##### Answer:

##### Question 101:

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

##### Answer:

##### Question 102:

- e
^{x}+ e^{–y}= c - e
^{x}+ e^{y}= c - e
^{–x}+ e^{y}= c - e
^{–x}+ e^{–y}= c

##### Answer:

e^{x} + e^{–y} = c

##### Question 103:

Which of the following differential equations cannot be solved, using variable separable method ?

- (y
^{2}– 2xy) dx = (x^{2}– 2xy) dy

##### Answer:

(y^{2} – 2xy) dx = (x^{2} – 2xy) dy

##### Question 104:

- y = vx
- v = yx
- x = vy
- x = v.

##### Answer:

x = vy

##### Question 105:

Which of the following is a homogeneous differential equation ?

- (4x + 6y + 5) dy – (3y + 2x + 4) dx = 0
- xy dx – (x
^{3}+ y^{3}) dy = 0 - (x
^{3}+ 2y^{2}) dx + 2xy dy = 0 - y
^{2}dx + (x^{2}– xy – y^{2}) dy = 0.

##### Answer:

y^{2} dx + (x^{2} – xy – y^{2}) dy = 0.

##### Question 106:

- e
^{–x} - e
^{–y} - x

##### Answer:

##### Question 107:

##### Answer:

##### Question 108:

- xy = c
- x = cy
^{2} - y = cx
- y = cx
^{2}

##### Answer:

y = cx

##### Question 109:

##### Answer:

##### Question 110:

The general solution of the differential equation ex dy + (y e^{x} + 2x) dx = 0 is :

- x e
^{y}+ x^{2}= c - x e
^{y}+ y^{2}= c - y e
^{x}+ x^{2}= c - y e
^{y}+ x^{2}= c.

##### Answer:

y e^{x} + x^{2} = c

##### Question 111:

The degree of the differential equation representing the family of curves (x – a)^{2} + y^{2} = 16 is :

- 0
- 2
- 3
- 1

##### Answer:

1