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

#### Chapters

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## Chapter 4: Differential Equations

Exercise 4.1Exercise 4.2Exercise 4.3Exercise 4.4Exercise 4.5Exercise 4.6Miscellaneous problems
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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

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Business Mathematics and Statistics Answers Guide chapter 4 (Differential Equations) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Tamil Nadu Board of Secondary Education Class 12th Business Mathematics and Statistics Answers Guide solutions in a manner that help students grasp basic concepts better and faster.

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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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