Tamil Nadu Board of Secondary EducationHSC Commerce Class 12th

# Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide chapter 4 - Differential Equations [Latest edition]

#### Chapters ## Chapter 4: Differential Equations

Exercise 4.1Exercise 4.2Exercise 4.3Exercise 4.4Exercise 4.5Exercise 4.6Miscellaneous problems
Exercise 4.1 [Pages 84 - 85]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 4 Differential EquationsExercise 4.1 [Pages 84 - 85]

Exercise 4.1 | Q 1. (i) | Page 84

Find the order and degree of the following differential equation:

("d"y)/("d"x) + 2 = x^3

Exercise 4.1 | Q 1. (ii) | Page 84

Find the order and degree of the following differential equation:

("d"^3y)/("d"x^3) + 3 (("d"y)/("d"x))^3 + 2 ("d"y)/("d"x) = 0

Exercise 4.1 | Q 1. (iii) | Page 85

Find the order and degree of the following differential equation:

("d"^2y)/("d"x^2) = sqrt(y - ("d"y)/("d"x))

Exercise 4.1 | Q 1. (iv) | Page 85

Find the order and degree of the following differential equation:

("d"^3y)/("d"x^3) = 0

Exercise 4.1 | Q 1. (v) | Page 85

Find the order and degree of the following differential equation:

("d"^2y)/("d"x^2) + y + (("d"y)/("d"x) - ("d"^3y)/("d"x^3))^(3/2) = 0

Exercise 4.1 | Q 1. (vi) | Page 85

Find the order and degree of the following diff erential equation:

(2 – y”)2 = y”2 + 2y’

Exercise 4.1 | Q 1. (vii) | Page 85

Find the order and degree of the following differential equation:

(("d"y)/("d"x))^3 + y = x -  ("d"x)/("d"y)

Exercise 4.1 | Q 2. (i) | Page 85

Find the differential equation of the following:

y = cx + c – c3

Exercise 4.1 | Q 2. (ii) | Page 85

Find the differential equation of the following:

y = c(x – c)2

Exercise 4.1 | Q 2. (iii) | Page 85

Find the differential equation of the following:

xy = c2

Exercise 4.1 | Q 2. (iv) | Page 85

Find the differential equation of the following:

x2 + y2 = a2

Exercise 4.1 | Q 3 | Page 85

Form the differential equation by eliminating α and β from (x – α)2 + (y – β)2 = r2

Exercise 4.1 | Q 4 | Page 85

Find the differential equation of the family of all straight lines passing through the origin

Exercise 4.1 | Q 5 | Page 85

Form the differential equation that represents all parabolas each of which has a latus rectum 4a and whose axes are parallel to the x-axis

Exercise 4.1 | Q 6 | Page 85

Find the differential equation of all circles passing through the origin and having their centers on the y axis

Exercise 4.1 | Q 7 | Page 85

Find the differential equation of the family of a parabola with foci at the origin and axis along the x-axis

Exercise 4.2 [Page 88]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 4 Differential EquationsExercise 4.2 [Page 88]

Exercise 4.2 | Q 1. (i) | Page 88

Solve: ("d"y)/("d"x) = "ae"^y

Exercise 4.2 | Q 1. (ii) | Page 88

Solve: (1 + x^2)/(1 + y) = xy ("d"y)/("d"x)

Exercise 4.2 | Q 2 | Page 88

Solve: y(1 - x) - x ("d"y)/("d"x) = 0

Exercise 4.2 | Q 3. (i) | Page 88

Solve: ydx – xdy = 0 dy

Exercise 4.2 | Q 3. (ii) | Page 88

Solve: ("d"y)/("d"x) + "e"^x + y"e"^x = 0

Exercise 4.2 | Q 4 | Page 88

Solve : cos x(1 + cosy) dx – sin y(1 + sinx) dy = 0

Exercise 4.2 | Q 5 | Page 88

Solve: (1 – x) dy – (1 + y) dx = 0

Exercise 4.2 | Q 6. (i) | Page 88

Solve: ("d"y)/("d"x) = y sin 2x

Exercise 4.2 | Q 6. (ii) | Page 88

Solve: log(("d"y)/("d"x)) = ax + by

Exercise 4.2 | Q 7 | Page 88

Find the curve whose gradient at any point P(x, y) on it is (x - "a")/(y - "b") and which passes through the origin

Exercise 4.3 [Page 92]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 4 Differential EquationsExercise 4.3 [Page 92]

Exercise 4.3 | Q 1 | Page 92

Solve the following homogeneous differential equation:

x ("d"y)/("d"x) = x + y

Exercise 4.3 | Q 2 | Page 92

Solve the following homogeneous differential equation:

(x - y) ("d"y)/("d"x) = x + 3y

Exercise 4.3 | Q 3 | Page 92

Solve the following homogeneous differential equation:

x ("d"y)/("d"x) - y = sqrt(x^2 + y^2)

Exercise 4.3 | Q 4 | Page 92

Solve the following homogeneous differential equation:

("d"y)/("d"x) = (3x - 2y)/(2x - 3y)

Exercise 4.3 | Q 5 | Page 92

Solve the following homogeneous differential equation:

(y2 – 2xy) dx = (x2 – 2xy) dy

Exercise 4.3 | Q 6 | Page 92

Solve the following homogeneous differential equation:

The slope of the tangent to a curve at any point (x, y) on it is given by (y3 – 2yx2) dx + (2xy2 – x3) dy = 0 and the curve passes through (1, 2). Find the equation of the curve

Exercise 4.3 | Q 7 | Page 92

Solve the following homogeneous differential equation:

An electric manufacturing company makes small household switches. The company estimates the marginal revenue function for these switches to be (x2 + y2) dy = xy dx where x represents the number of units (in thousands). What is the total revenue function?

Exercise 4.4 [Pages 94 - 95]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 4 Differential EquationsExercise 4.4 [Pages 94 - 95]

Exercise 4.4 | Q 1 | Page 94

Solve the following:

("d"y)/("d"x) - y/x = x

Exercise 4.4 | Q 2 | Page 94

Solve the following:

("d"y)/(""dx) + y cos x = sin x cos x

Exercise 4.4 | Q 3 | Page 94

Solve the following:

x ("d"y)/("d"x) + 2y = x^4

Exercise 4.4 | Q 4 | Page 94

Solve the following:

("d"y)/("d"x) + (3x^2)/(1 + x^3) y = (1 + x^2)/(1 + x^3)

Exercise 4.4 | Q 5 | Page 94

Solve the following:

("d"y)/("d"x) + y/x = x'e"^x

Exercise 4.4 | Q 6 | Page 94

Solve the following:

("d"y)/("d"x) + y tan x = cos^3x

Exercise 4.4 | Q 7 | Page 94

Solve the following:

If ("d"y)/("d"x) + 2 y tan x = sin x and if y = 0 when x = pi/3 express y in term of x

Exercise 4.4 | Q 8 | Page 95

Solve the following:

("d"y)/("d"x) + y/x = x"e"^x

Exercise 4.4 | Q 9 | Page 95

Solve the following:

A bank pays interest by continuous compounding, that is by treating the interest rate as the instantaneous rate of change of principal. A man invests ₹ 1,00,000 in the bank deposit which accrues interest, 8% per year compounded continuously. How much will he get after 10 years? (e0.8 = 2.2255)

Exercise 4.5 [Pages 98 - 99]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 4 Differential EquationsExercise 4.5 [Pages 98 - 99]

Exercise 4.5 | Q 1 | Page 98

Solve the following differential equation:

("d"^2y)/("d"x^2) - 6 ("d"y)/("d"x) + 8y = 0

Exercise 4.5 | Q 2 | Page 98

Solve the following differential equation:

("d"^2y)/("d"x^2) - 4("d"y)/("d"x) + 4y = 0

Exercise 4.5 | Q 3 | Page 99

Solve the following differential equation:

(D2 + 2D + 3)y = 0

Exercise 4.5 | Q 4 | Page 99

Solve the following differential equation:

("d"^2y)/("d"x^2) - 2"k" ("d"y)/("d"x) + "k"^2y = 0

Exercise 4.5 | Q 5 | Page 99

Solve the following differential equation:

(D2 – 2D – 15)y = 0 given that ("d"y)/("d"x) = 0 and ("d"^2y)/("d"x^2) = 2 when x = 0

Exercise 4.5 | Q 6 | Page 99

Solve the following differential equation:

(4D2 + 4D – 3)y = e2x

Exercise 4.5 | Q 7 | Page 99

Solve the following differential equation:

("d"^2y)/("d"x^2) + 16y = 0

Exercise 4.5 | Q 8 | Page 99

Solve the following differential equation:

(D2 – 3D + 2)y = e3x which shall vanish for x = 0 and for x = log 2

Exercise 4.5 | Q 9 | Page 99

Solve the following differential equation:

(D2 + D – 2)y = e3x + e–3x

Exercise 4.5 | Q 10 | Page 99

Solve the following differential equation:

(D2 – 10D + 25) y = 4e5x + 5

Exercise 4.5 | Q 11 | Page 99

Solve the following differential equation:

(4"D"^2 + 16"D" + 15)y = 4"e"^((-3)/2x)

Exercise 4.5 | Q 12 | Page 99

Solve the following differential equation:

(3D2 + D – 14)y – 13e2x

Exercise 4.5 | Q 13 | Page 99

Solve the following differential equation:

Suppose that the quantity demanded Qd = 13 - 6"p" + 2 "dp"/"dt" + ("d"^2"p")/("dt"^2) and quantity supplied Qs = - 3 + 2"p" where p is the price. Find the equilibrium price for market clearance

Exercise 4.6 [Pages 99 - 101]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 4 Differential EquationsExercise 4.6 [Pages 99 - 101]

#### MCQ

Exercise 4.6 | Q 1 | Page 99

Choose the correct alternative:

The degree of the differential equation ("d"^4y)/("d"x^4) - (("d"^2y)/("d"x^2))^4 + ("d"y)/("d"x) = 3

• 1

• 2

• 3

• 4

Exercise 4.6 | Q 2 | Page 99

Choose the correct alternative:

The order and degree of the differential equation sqrt(("d"^2y)/("d"x^2)) = sqrt(("d"y)/("d"x) + 5) are respectively

• 2 and 3

• 3 and 2

• 2 and 1

• 2 and 2

Exercise 4.6 | Q 3 | Page 99

Choose the correct alternative:

The order and degree of the differential equation (("d"^2y)/("d"x^2))^(3/2) - sqrt((("d"y)/("d"x))) - 4 = 0 are respectively

• 2 and 6

• 3 and 6

• 1 and 4

• 2 and 4

Exercise 4.6 | Q 4 | Page 99

Choose the correct alternative:

Th e differential equation (("d"x)/("d"y))^3 + 2y^(1/2) = x is

• of order 2 and degree 1

• of order 1 and degree 3

• of order 1 and degree 6

• of order 1 and degree 2

Exercise 4.6 | Q 5 | Page 99

Choose the correct alternative:

The differential equation formed by eliminating a and b from y = aex + be-x is

• ("d"^2y)/("d"x^2) - y = 0

• ("d"^2y)/("d"x^2) - ("d"y)/("d"x) = 0

• ("d"^2y)/("d"x^2) = 0

• ("d"^2y)/("d"x^2) - x = 0

Exercise 4.6 | Q 6 | Page 99

Choose the correct alternative:

If y = ex + c – c3 then its differential equation is

• y = x ("d"y)/("d"x) + ("d"y)/("d"x) - (("d"y)/("d'x))^3

• y + (("d"y)/("d"x))^3 = x ("d"y)/("d"x) - ("d"y)/("d"x)

• ("d"y)/("d"x) + y = (("d"y)/("d"x))^3 - x ("d"y)/("d"x)

• ("d"^3y)/("d"x^3) = 0

Exercise 4.6 | Q 7 | Page 99

Choose the correct alternative:

The integrating factor of the differential equation ("d"y)/("d"x) + "P"x = Q is

• "e"^(int Pdx)

• int Pdx

• int Pdy

• "e"^(int Pdy)

Exercise 4.6 | Q 8 | Page 99

Choose the correct alternative:

The complementary function of (D2 + 4) y = e2x is

• (Ax + B)e2x

• (Ax + B)e-2x

• A cos2x + B sin2x

• Ae-2x + Be2x

Exercise 4.6 | Q 9 | Page 100

Choose the correct alternative:

The differential equation of y = mx + c is (m and c are arbitrary constants)

• ("d"^2y)/("d"x^2) = 0

• y = x ("d"y)/("d"x) + "c"

• xdy + ydx = 0

• yd – xdy = 0

Exercise 4.6 | Q 10 | Page 100

Choose the correct alternative:

The particular intergral of the differential equation ("d"^2y)/("d"x^2) - 8 ("d"y)/("d"x) + 16y = 2"e"^(4x)

• (x^2"e"^(4x))/(2!)

• "e"^(4x)/(2!)

• x^2"e"^(4x)

• x"e"^(4x)

Exercise 4.6 | Q 11 | Page 100

Choose the correct alternative:

Solution of ("d"x)/("d"y) + "P"x = 0

• x = cepy

• x = ce-py

• x = py + c

• x = cy

Exercise 4.6 | Q 12 | Page 100

Choose the correct alternative:

If sec2 x is an integrating factor of the differential equation ("d"y)/("d"x) + "P"y = Q then P =

• 2 tan x

• sec x

• cos2x

• tan2x

Exercise 4.6 | Q 13 | Page 100

Choose the correct alternative:

The integrating factor of x ("d"y)/("d"x) - y = x^2 is

• (-1)/x

• 1/x

• log x

• x

Exercise 4.6 | Q 14 | Page 100

Choose the correct alternative:

The solution of the differential equation ("d"y)/("d"x) + "P"y = Q where P and Q are the function of x is

• y = int "Qe"^(int pdx)  "d"x + "c"

• y = int "Qe"^(-int pdx)  "d"x + "c"

• y"e"^(intpdx) = int "Qe"^(intPdx)  "d"x + "c"

• y"e"^(intpdx) = int "Qe"^(- intPdx)  "d"x + "c"

Exercise 4.6 | Q 15 | Page 100

Choose the correct alternative:

The differential equation formed by eliminating A and B from y = e-2x (A cos x + B sin x) is

• y2 – 4y1 + 5 = 0

• y2 + 4y – 5 = 0

• y2 – 4y1 + 5 = 0

• y2 + 4y1 – 5 = 0

Exercise 4.6 | Q 16 | Page 100

Choose the correct alternative:

The particular integral of the differential equation f(D) y = eax where f(D) = (D – a)

• x^2/2 "e"^(ax)

• x"e"^(ax)

• x/2 "e"^(ax)

• x^2"e"^(ax)

Exercise 4.6 | Q 17 | Page 100

Choose the correct alternative:

The differential equation of x2 + y2 = a2

• xdy + ydx = 0

• ydx – xdy = 0

• ydx – xdy = 0

• xdx + ydy = 0

Exercise 4.6 | Q 18 | Page 100

Choose the correct alternative:

The complementary function of ("d"^2y)/("d"x^2) - ("d"y)/("d"x) = 0 is

• A + Bex

• (A + B)ex

• (Ax + B)ex

• Aex + B

Exercise 4.6 | Q 19 | Page 100

Choose the correct alternative:

The P.I of (3D2 + D – 14)y = 13e2x is

• 1/2 "e"^x

• xe2x

• x^2/2 "e"^(2x)

• Aex + B

Exercise 4.6 | Q 20 | Page 100

Choose the correct alternative:

The general solution of the differential equation ("d"y)/("d"x) = cos x is

• y = sin x + 1

• y = sin x – 2

• y = cos x + c, c is an arbitrary constant

• y = sin x + c, c is an arbitrary constant

Exercise 4.6 | Q 21 | Page 101

Choose the correct alternative:

A homogeneous differential equation of the form ("d"y)/("d"x) = f(y/x) can be solved by making substitution

• y = vx

• y = yx

• x = vy

• x = v

Exercise 4.6 | Q 22 | Page 101

Choose the correct alternative:

A homogeneous differential equation of the form ("d"x)/("d"y) = f(x/y) can be solved by making substitution

• x = vy

• y = vx

• y = v

• x = v

Exercise 4.6 | Q 23 | Page 101

Choose the correct alternative:

The variable separable form of ("d"y)/("d"x) = (y(x - y))/(x(x + y)) by taking y = vx and ("d"y)/("d"x) = "v" + x "dv"/("d"x) is

• (2"v"^2)/(1 +"v")  "dv" = ("d"x)/x

• (2"v"^2)/(1 +"v")  "dv" = - ("d"x)/x

• (2"v"^2)/(1 - "v")  "dv" = ("d"x)/x

• (1 + "v")/(2"v"^2) "dv" = - ("d"x)/x

Exercise 4.6 | Q 24 | Page 101

Choose the correct alternative:

Which of the following is the homogeneous differential equation?

• (3x – 5) dx = (4y – 1) dy

• xy dx – (x3 + y3) dy = 0

• y2 dx + (x2 – xy – y2) dy = 0

• (x2 + y) dx = (y2 + x) dy

Exercise 4.6 | Q 25 | Page 101

Choose the correct alternative:

The solution of the differential equation ("d"y)/("d"x) = y/x + (f(y/x))/(f"'"(y/x)) is

• f(y/x) = kx

• x f(y/x) = k

• f(y/x) = ky

• yf(y/x) = k

Miscellaneous problems [Page 101]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide Chapter 4 Differential EquationsMiscellaneous problems [Page 101]

Miscellaneous problems | Q 1 | Page 101

Suppose that Qd = 30 - 5"P" + 2 "dP"/"dt" + ("d"^2"P")/("dt"^2) and Qs = 6 + 3P. Find the equilibrium price for market clearance

Miscellaneous problems | Q 2 | Page 101

Form the differential equation having for its general solution y = ax2 + bx

Miscellaneous problems | Q 3 | Page 101

Solve yx2dx + e–x dy = 0

Miscellaneous problems | Q 4 | Page 101

Solve (x2 + y2) dx + 2xy dy = 0

Miscellaneous problems | Q 5 | Page 101

Solve x ("d"y)/(d"x) + 2y = x^4

Miscellaneous problems | Q 6 | Page 101

A manufacturing company has found that the cost C of operating and maintaining the equipment is related to the length ’m’ of intervals between overhauls by the equation "m"^2 "dC"/"dm" + 2"mC" = 2 and c = 4 and when  = 2. Find the relationship between C and m

Miscellaneous problems | Q 7 | Page 101

Solve (D2 – 3D + 2)y = e4x given y = 0 when x = 0 and x = 1

Miscellaneous problems | Q 8 | Page 101

Solve ("d"y)/("d"x) + y cos x + x = 2 cos x

Miscellaneous problems | Q 9 | Page 101

Solve x2ydx – (x3 + y3) dy = 0

Miscellaneous problems | Q 10 | Page 101

Solve ("d"y)/("d"x) = xy + x + y + 1

## Chapter 4: Differential Equations

Exercise 4.1Exercise 4.2Exercise 4.3Exercise 4.4Exercise 4.5Exercise 4.6Miscellaneous problems ## Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide chapter 4 - Differential Equations

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Concepts covered in Class 12th Business Mathematics and Statistics Answers Guide chapter 4 Differential Equations are Formation of Ordinary Differential Equations, Solution of First Order and First Degree Differential Equations, Second Order First Degree Differential Equations with Constant Coefficients.

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