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## Chapter 8: Differential Equation and Applications

### 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]

Determine the order and degree of the following differential equations.

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

Determine the order and degree of the following differential equations.

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

Determine the order and degree of the following differential equations.

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

Determine the order and degree of the following differential equations.

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

Determine the order and degree of the following differential equations.

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

Determine the order and degree of the following differential equations.

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

Determine the order and degree of the following differential equations.

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

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 |

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` |

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

Solution |
D.E. |

y = e^{x} |
`dy/ dx= y` |

Determine the order and degree of the following differential equations.

Solution |
D.E. |

y = 1 − logx | `x^2(d^2y)/dx^2 = 1` |

Determine the order and degree of the following differential equations.

Solution |
D.E |

y = ae^{x }+ be^{−x} |
`(d^2y)/dx^2= 1` |

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` |

### 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]

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

y = Ae^{3x} + Be^{−3x}

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

`y = c_2 + c_1/x`

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

y = (c_{1} + c_{2} x) e^{x}

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

y = c_{1}e^{ 3x} + c_{2}e ^{2x}

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

y^{2} = (x + c)^{3}

Find the differential equation by eliminating arbitrary constants from the relation x^{2} + y^{2 }= 2ax

Form the differential equation by eliminating arbitrary constants from the relation

bx + ay = ab.

Find the differential equation whose general solution is

x^{3} + y^{3} = 35ax.

Form the differential equation from the relation x^{2 }+ 4y^{2 }= 4b^{2}

### 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]

Solve the following differential equation.

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

Solve the following differential equation.

`(dθ)/dt = − k (θ − θ_0)`

Solve the following differential equation

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

Solve the following differential equation.

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

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

(x − y^{2} x) dx − (y + x^{2} y) dy = 0, when x = 2, y = 0

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

`(x + 1) dy/dx − 1 = 2e^(−y)` ,

when y = 0, x = 1

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

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

when x=e, y = e^{2}.

For the following differential equation find the particular solution.

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

when y = 1, x = 0

### 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]

Solve the following differential equation.

xdx + 2y dy = 0

Solve the following differential equation.

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

Solve the following differential equation.

x^{2}y dx − (x^{3} + y^{3} ) dy = 0

**Solve the following differential equation.**

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

**Solve the following differential equation.**

(x^{2} − y^{2 }) dx + 2xy dy = 0

**Solve the following differential equation.**

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

Solve the following differential equation.

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

### 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]

Solve the following differential equation.

`dy/dx + y = e ^-x`

Solve the following differential equation.

`dy/dx +y =3`

Solve the following differential equation.

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

Solve the following differential equation.

`(x + y) dy/dx = 1`

Solve the following differential equation.

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

Solve the following differential equation.

`dy/dx + 2xy = x`

Solve the following differential equation.

`(x + a) dy/dx = – y + a`

Solve the following differential equation.

dr + (2r)dθ= 8dθ

### 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]

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.

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)`

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`)

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.

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.

### 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]

**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

**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

**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`

**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`

**Choose the correct alternative.**

The solution of `dy/ dx` = 1 is

x + y = c

xy = c

x

^{2}+ y^{2 }= cy − x = c

**Choose the correct alternative.**

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

x

^{3}+ y^{3}= 7x

^{2}+ y^{2}= cx

^{3}+ y^{3}= cx + y = c

**Choose the correct alternative.**

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

y = ae

^{x}y = be

^{2x}y = be

^{-2x}y = e

^{ax}

**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

**Choose the correct alternative.**

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

x

– x

e

e

^{–x}

**Choose the correct alternative.**

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

ye

^{−x }= x + cye

^{x}= x + cye

^{x}= 2x + cye

^{−x}= 2x + c

**Fill in the blank:**

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

**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.

**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.

**Fill in the blank:**

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

**Fill in the blank:**

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

**Fill in the blank:**

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

**State whether the following is True or False:**

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

True

False

**State whether the following is True or False:**

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

True

False

**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

**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

**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

**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

**Find the order and degree of the following differential equation:**

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

**Find the order and degree of the following differential equation:**

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

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

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

**Solve the differential equation:**

`dy/dx = 1 +x+ y + xy`

**Solve the differential equation:**

`e^(dy/dx) = x`

**Solve the differential equation:**

dr = a r dθ − θ dr

**Solve the differential equation:**

Find the differential equation of family of curves y = e^{x} (ax + bx^{2}), where A and B are arbitrary constants.

**Solve **

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

**Solve **

**y dx – x dy = −log x dx**

**Solve **

**`y log y dy/dx + x – log y = 0`**

**Solve:**

**(x + y) dy = a ^{2 }dx**

**Solve **

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

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?

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?

**y2 dx + (xy + x ^{2})dy = 0**

**x ^{2}y dx – (x^{3} + y^{3}) dy = 0**

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

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

y dx – x dy + log x dx = 0

** `dy/dx = log x`**

**Solve**

`y log y dx/ dy = log y – x`

## Chapter 8: Differential Equation and Applications

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

Balbharati solutions for Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board chapter 8 (Differential Equation and Applications) 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 Maharashtra State Board Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

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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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