# Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 8 - Differential Equation and Applications [Latest edition]

## Chapter 8: Differential Equation and Applications

Exercise 8.1Exercise 8.2Exercise 8.3Exercise 8.4Exercise 8.5Exercise 8.6Miscellaneous Exercise 8
Exercise 8.1 [Page 162]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 8 Differential Equation and Applications Exercise 8.1 [Page 162]

Exercise 8.1 | Q 1.1 | Page 162

Determine the order and degree of the following differential equations.

(d^2x)/(dt^2)+((dx)/(dt))^2 + 8=0

Exercise 8.1 | Q 1.2 | Page 162

Determine the order and degree of the following differential equations.

((d^2y)/(dx^2))^2 + ((dy)/(dx))^2 =a^x

Exercise 8.1 | Q 1.3 | Page 162

Determine the order and degree of the following differential equations.

(d^4y)/dx^4 + [1+(dy/dx)^2]^3 = 0

Exercise 8.1 | Q 1.4 | Page 162

Determine the order and degree of the following differential equations.

(y''')^2 + 2(y'')^2 + 6y' + 7y = 0

Exercise 8.1 | Q 1.5 | Page 162

Determine the order and degree of the following differential equations.

sqrt(1+1/(dy/dx)^2) = (dy/dx)^(3/2)

Exercise 8.1 | Q 1.6 | Page 162

Determine the order and degree of the following differential equations.

dy/dx = 7 (d^2y)/dx^2

Exercise 8.1 | Q 1.7 | Page 162

Determine the order and degree of the following differential equations.

((d^3y)/dx^3)^(1/6) = 9

Exercise 8.1 | Q 2.1 | Page 162

In each of the following examples, verify that the given function is a solution of the corresponding differential equation.

 Solution D.E. xy = log y +k y' (1-xy) =y2
Exercise 8.1 | Q 2.2 | Page 162

In the following example, verify that the given function is a solution of the corresponding differential equation.

 Solution D.E. y = xn x^2(d^2y)/dx^2 - n xx (xdy)/dx + ny =0
Exercise 8.1 | Q 2.3 | Page 162

In each of the following examples, verify that the given function is a solution of the corresponding differential equation.

 Solution D.E. y = ex dy/ dx= y
Exercise 8.1 | Q 2.4 | Page 162

Determine the order and degree of the following differential equations.

 Solution D.E. y = 1 − logx x^2(d^2y)/dx^2 = 1
Exercise 8.1 | Q 2.5 | Page 162

Determine the order and degree of the following differential equations.

 Solution D.E y = aex + be−x (d^2y)/dx^2= 1
Exercise 8.1 | Q 2.6 | Page 162

Determine the order and degree of the following differential equations.

 Solution D.E. ax2 + by2 = 5 xy(d^2y)/dx^2+ x(dy/dx)^2 = y dy/dx
Exercise 8.2 [Page 163]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 8 Differential Equation and Applications Exercise 8.2 [Page 163]

Exercise 8.2 | Q 1.1 | Page 163

Obtain the differential equation by eliminating arbitrary constants from the following equations.

y = Ae3x + Be−3x

Exercise 8.2 | Q 1.2 | Page 163

Obtain the differential equations by eliminating arbitrary constants from the following equation.

y = c_2 + c_1/x

Exercise 8.2 | Q 1.3 | Page 163

Obtain the differential equation by eliminating arbitrary constants from the following equations.

y = (c1 + c2 x) ex

Exercise 8.2 | Q 1.4 | Page 163

Obtain the differential equations by eliminating arbitrary constants from the following equations.

y = c1e 3x + c2e 2x

Exercise 8.2 | Q 1.5 | Page 163

Obtain the differential equation by eliminating arbitrary constants from the following equations.

y2 = (x + c)3

Exercise 8.2 | Q 2 | Page 163

Find the differential equation by eliminating arbitrary constants from the relation x2 + y2 = 2ax

Exercise 8.2 | Q 3 | Page 163

Form the differential equation by eliminating arbitrary constants from the relation

bx + ay = ab.

Exercise 8.2 | Q 4 | Page 163

Find the differential equation whose general solution is

x3 + y3 = 35ax.

Exercise 8.2 | Q 5 | Page 163

Form the differential equation from the relation x2 + 4y2 = 4b2

Exercise 8.3 [Page 165]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 8 Differential Equation and Applications Exercise 8.3 [Page 165]

Exercise 8.3 | Q 1.1 | Page 165

Solve the following differential equation.

dy/dx = x^2 y + y

Exercise 8.3 | Q 1.2 | Page 165

Solve the following differential equation.

(dθ)/dt  = − k (θ − θ_0)

Exercise 8.3 | Q 1.3 | Page 165

Solve the following differential equation

(x2 − yx2 ) dy + (y2 + xy2) dx = 0

Exercise 8.3 | Q 1.4 | Page 165

Solve the following differential equation.

y^3 - dy/dx = x dy/dx

Exercise 8.3 | Q 2.1 | Page 165

For each of the following differential equations find the particular solution.

(x − y2 x) dx − (y + x2 y) dy = 0, when x = 2, y = 0

Exercise 8.3 | Q 2.2 | Page 165

For each of the following differential equations find the particular solution.

(x + 1) dy/dx − 1 = 2e^(−y) ,

when  y = 0, x = 1

Exercise 8.3 | Q 2.3 | Page 165

For each of the following differential equations find the particular solution.

y (1 + logx)dx/dy - x log x = 0,

when x=e, y = e2.

Exercise 8.3 | Q 2.4 | Page 165

For  the following differential equation find the particular solution.

dy/ dx = (4x + y + 1),

when  y = 1, x = 0

Exercise 8.4 [Page 167]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 8 Differential Equation and Applications Exercise 8.4 [Page 167]

Exercise 8.4 | Q 1.1 | Page 167

Solve the following differential equation.

xdx + 2y dy = 0

Exercise 8.4 | Q 1.2 | Page 167

Solve the following differential equation.

y2 dx + (xy + x2 ) dy = 0

Exercise 8.4 | Q 1.3 | Page 167

Solve the following differential equation.

x2y dx − (x3 + y3 ) dy = 0

Exercise 8.4 | Q 1.4 | Page 167

Solve the following differential equation.

dy /dx +(x-2 y)/ (2x- y)= 0

Exercise 8.4 | Q 1.5 | Page 167

Solve the following differential equation.

(x2 − y2 ) dx + 2xy dy = 0

Exercise 8.4 | Q 1.6 | Page 167

Solve the following differential equation.

xy  dy/dx = x^2 + 2y^2

Exercise 8.4 | Q 1.7 | Page 167

Solve the following differential equation.

x^2 dy/dx = x^2 +xy - y^2

Exercise 8.5 [Page 168]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 8 Differential Equation and Applications Exercise 8.5 [Page 168]

Exercise 8.5 | Q 1.1 | Page 168

Solve the following differential equation.

dy/dx + y = e ^-x

Exercise 8.5 | Q 1.2 | Page 168

Solve the following differential equation.

dy/dx +y =3

Exercise 8.5 | Q 1.3 | Page 168

Solve the following differential equation.

x dy/dx + 2y = x^2 log x

Exercise 8.5 | Q 1.4 | Page 168

Solve the following differential equation.

(x + y) dy/dx = 1

Exercise 8.5 | Q 1.5 | Page 168

Solve the following differential equation.

y dx + (x - y2 ) dy = 0

Exercise 8.5 | Q 1.6 | Page 168

Solve the following differential equation.

dy/dx + 2xy = x

Exercise 8.5 | Q 1.7 | Page 168

Solve the following differential equation.

(x + a) dy/dx = – y + a

Exercise 8.5 | Q 1.8 | Page 168

Solve the following differential equation.

dr + (2r)dθ= 8dθ

Exercise 8.6 [Page 170]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 8 Differential Equation and Applications Exercise 8.6 [Page 170]

Exercise 8.6 | Q 1 | Page 170

In a certain culture of bacteria, the rate of increase is proportional to the number present. If it is found that the number doubles in 4 hours, find the number of times the bacteria are increased in 12 hours.

Exercise 8.6 | Q 2 | Page 170

The population of a town increases at a rate proportional to the population at that time. If the population increases from 40 thousands to 60 thousands in 40 years, what will be the population in another 20 years?

(Given: sqrt(3/2)= 1.2247)

Exercise 8.6 | Q 3 | Page 170

The rate of growth of bacteria is proportional to the number present. If initially, there were 1000 bacteria and the number doubles in 1 hour, find the number of bacteria after 5/2 hours.

(Given: sqrt2 = 1.414)

Exercise 8.6 | Q 4 | Page 170

Find the population of a city at any time t, given that the rate of increase of population is proportional to the population at that instant and that in a period of 40 years, the population increased from 30,000 to 40,000.

Exercise 8.6 | Q 5 | Page 170

The rate of depreciation (dV)/ dt of a machine is inversely proportional to the square of t + 1, where V is the value of the machine t years after it was purchased. The initial value of the machine was ₹ 8,00,000 and its value decreased ₹1,00,000 in the first year. Find its value after 6 years.

Miscellaneous Exercise 8 [Pages 171 - 173]

### Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 8 Differential Equation and Applications Miscellaneous Exercise 8 [Pages 171 - 173]

Miscellaneous Exercise 8 | Q 1.01 | Page 171

Choose the correct alternative.

The order and degree of (dy/dx)^3 - (d^3y)/dx^3 + ye^x = 0 are respectively.

• 3, 1

• 1, 3

• 3, 3

• 1, 1

Miscellaneous Exercise 8 | Q 1.02 | Page 171

Choose the correct alternative.

The order and degree of [ 1+ (dy/dx)^3]^(2/3) = 8 (d^3y)/dx^3 are respectively.

• 3, 1

• 1, 3

• 3, 3

• 1, 1

Miscellaneous Exercise 8 | Q 1.03 | Page 171

Choose the correct alternative.

The differential equation of y = k_1 + k_2/x is

• (d^2y)/dx^2 + 2 dy/dx = 0

• x(d^2y)/dx^2 + 2 dy/dx = 0

• (d^2y)/dx^2 -2 dy/dx = 0

• x(d^2y)/dx^2 -2 dy/dx = 0

Miscellaneous Exercise 8 | Q 1.04 | Page 171

Choose the correct alternative.

The differential equation of y = k_1e^x+ k_2 e^-x is

• (d^2y)/dx^2 - y = 0

• (d^2y)/dx^2 + dy/dx  = 0

• (d^2y)/dx^2 + ydy/dx  = 0*

• (d^2y)/dx^2 + y  = 0

Miscellaneous Exercise 8 | Q 1.05 | Page 171

Choose the correct alternative.

The solution of dy/ dx = 1 is

• x + y = c

• xy = c

• x2 + y2 = c

• y − x = c

Miscellaneous Exercise 8 | Q 1.06 | Page 171

Choose the correct alternative.

The solution of dy/dx + x^2/y^2 = 0 is

• x3 + y3 = 7

• x2 + y2 = c

• x3 + y3 = c

• x + y = c

Miscellaneous Exercise 8 | Q 1.07 | Page 172

Choose the correct alternative.

The solution of x dy/dx = y log y is

• y = aex

• y = be2x

• y = be-2x

• y = eax

Miscellaneous Exercise 8 | Q 1.08 | Page 172

Choose the correct alternative.

Bacteria increases at the rate proportional to the number present. If the original number M doubles in 3 hours, then the number of bacteria will be 4M in

• 4 hours

• 6 hours

• 8 hours

• 10 hours

Miscellaneous Exercise 8 | Q 1.09 | Page 172

Choose the correct alternative.

The integrating factor of dy/dx + y = e^-x

• x

• – x

• e

• e –x

Miscellaneous Exercise 8 | Q 1.1 | Page 172

Choose the correct alternative.

The integrating factor of dy/dx -  y = e^x is ex, then its solution is

• ye −x = x + c

• yex = x + c

• yex = 2x + c

• ye−x = 2x + c

Miscellaneous Exercise 8 | Q 2.1 | Page 172

Fill in the blank:

The order of highest derivative occurring in the differential equation is called ___________ of the differential equation.

Miscellaneous Exercise 8 | Q 2.2 | Page 172

Fill in the blank:

The power of the highest ordered derivative when all the derivatives are made free from negative and / or fractional indices if any is called __________ of the differential equation.

Miscellaneous Exercise 8 | Q 2.3 | Page 172

Fill in the blank:

A solution of differential equation which can be obtained from the general solution by giving particular values to the arbitrary constants is called ___________ solution.

Miscellaneous Exercise 8 | Q 2.4 | Page 172

Fill in the blank:

Order and degree of a differential equation are always __________ integers.

Miscellaneous Exercise 8 | Q 2.5 | Page 172

Fill in the blank:

The integrating factor of the differential equation dy/dx – y = x is __________

Miscellaneous Exercise 8 | Q 2.6 | Page 172

Fill in the blank:

The differential equation by eliminating arbitrary constants from bx + ay = ab is __________.

Miscellaneous Exercise 8 | Q 3.1 | Page 172

State whether the following is True or False:

The integrating factor of the differential equation dy/dx - y = x is e-x

• True

• False

Miscellaneous Exercise 8 | Q 3.2 | Page 172

State whether the following is True or False:

Order and degree of a differential equation are always positive integers.

• True

• False

Miscellaneous Exercise 8 | Q 3.3 | Page 172

State whether the following is True or False:

The degree of a differential equation is the power of the highest ordered derivative when all the derivatives are made free from negative and/or fractional indices if any.

• True

• False

Miscellaneous Exercise 8 | Q 3.4 | Page 172

State whether the following is True or False:

The order of highest derivative occurring in the differential equation is called degree of the differential equation.

• True

• False

Miscellaneous Exercise 8 | Q 3.5 | Page 172

State whether the following is True or False:

The power of the highest ordered derivative when all the derivatives are made free from negative and / or fractional indices if any is called order of the differential equation.

• True

• False

Miscellaneous Exercise 8 | Q 3.6 | Page 172

State whether the following is True or False:

The degree of the differential equation e^((dy)/(dx)) = dy/dx +c is not defined.

• True

• False

Miscellaneous Exercise 8 | Q 4.01 | Page 172

Find the order and degree of the following differential equation:

[ (d^3y)/dx^3 + x]^(3/2) = (d^2y)/dx^2

Miscellaneous Exercise 8 | Q 4.01 | Page 172

Find the order and degree of the following differential equation:

x+ dy/dx = 1 + (dy/dx)^2

Miscellaneous Exercise 8 | Q 4.02 | Page 172

Verify y = log x + c is a solution of the differential equation

x(d^2y)/dx^2 + dy/dx = 0

Miscellaneous Exercise 8 | Q 4.03 | Page 172

Solve the differential equation:

dy/dx = 1 +x+ y + xy

Miscellaneous Exercise 8 | Q 4.03 | Page 172

Solve the differential equation:

e^(dy/dx) = x

Miscellaneous Exercise 8 | Q 4.03 | Page 173

Solve the differential equation:

dr = a r dθ − θ dr

Miscellaneous Exercise 8 | Q 4.03 | Page 173

Solve the differential equation:

Find the differential equation of family of curves y = ex (ax + bx2), where A and B are arbitrary constants.

Miscellaneous Exercise 8 | Q 4.04 | Page 173

Solve

dy/dx = (x+y+1)/(x+y-1)  when  x = 2/3 and y = 1/3

Miscellaneous Exercise 8 | Q 4.05 | Page 173

Solve

y dx – x dy = −log x dx

Miscellaneous Exercise 8 | Q 4.06 | Page 173

Solve

y log  y dy/dx + x  – log y = 0

Miscellaneous Exercise 8 | Q 4.07 | Page 173

Solve:

(x + y) dy = a2 dx

Miscellaneous Exercise 8 | Q 4.08 | Page 173

Solve

dy/dx + 2/ x y = x^2

Miscellaneous Exercise 8 | Q 4.09 | Page 173

The rate of growth of population is proportional to the number present. If the population doubled in the last 25 years and the present population is 1 lac, when will the city have population 4,00,000?

Miscellaneous Exercise 8 | Q 4.1 | Page 173

The resale value of a machine decreases over a 10 year period at a rate that depends on the age of the machine. When the machine is x years old, the rate at which its value is changing is ₹ 2200 (x − 10) per year. Express the value of the machine as a function of its age and initial value. If the machine was originally worth ₹1,20,000, how much will it be worth when it is 10 years old?

Miscellaneous Exercise 8 | Q 4.11 | Page 173

y2 dx + (xy + x2)dy = 0

Miscellaneous Exercise 8 | Q 4.12 | Page 173

x2y dx – (x3 + y3) dy = 0

Miscellaneous Exercise 8 | Q 4.13 | Page 173

xy dy/dx  = x^2 + 2y^2

Miscellaneous Exercise 8 | Q 4.14 | Page 173

(x + 2y^3 ) dy/dx = y

Miscellaneous Exercise 8 | Q 4.15 | Page 173

y dx – x dy + log x dx = 0

Miscellaneous Exercise 8 | Q 4.16 | Page 173

dy/dx = log x

Miscellaneous Exercise 8 | Q 4.17 | Page 173

Solve

y log y  dx/ dy = log y  – x`

## Chapter 8: Differential Equation and Applications

Exercise 8.1Exercise 8.2Exercise 8.3Exercise 8.4Exercise 8.5Exercise 8.6Miscellaneous Exercise 8

## Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 8 - Differential Equation and Applications

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Concepts covered in Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 8 Differential Equation and Applications are Differential Equations, Order and Degree of a Differential Equation, Formation of Differential Equation by Eliminating Arbitary Constant, Differential Equations with Variables Separable Method, Homogeneous Differential Equations, Linear Differential Equations, Application of Differential Equations.

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