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Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 6 - Differential Equations [Latest edition]

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Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 6 - Differential Equations - Shaalaa.com
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Chapter 6: Differential Equations

Exercise 6.1Exercise 6.2Exercise 6.3Exercise 6.4Exercise 6.5Exercise 6.6Miscellaneous exercise 1Miscellaneous exercise 2
Exercise 6.1 [Page 193]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 6 Differential Equations Exercise 6.1 [Page 193]

Exercise 6.1 | Q 1 | Page 193

Determine the order and degree of the following differential equation:

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

Exercise 6.1 | Q 2 | Page 193

Determine the order and degree of the following differential equation:

`root(3)(1 +("dy"/"dx")^2) = ("d"^2"y")/"dx"^2`

Exercise 6.1 | Q 3 | Page 193

Determine the order and degree of the following differential equation:

`"dy"/"dx" = (2 "sin x" + 3)/("dy"/"dx")`

Exercise 6.1 | Q 4 | Page 193

Determine the order and degree of the following differential equation:

`("d"^2"y")/"dx"^2 + "dy"/"dx" + "x" = sqrt(1 + ("d"^3"y")/"dx"^3)`

Exercise 6.1 | Q 5 | Page 193

Determine the order and degree of the following differential equation:

`("d"^2"y")/"dx"^2 + ("dy"/"dx")^2 + 7"x" + 5 = 0`

Exercise 6.1 | Q 6 | Page 193

Determine the order and degree of the following differential equation:

`("y"''')^2 + 3"y"'' + 3"xy"' + 5"y" = 0`

Exercise 6.1 | Q 7 | Page 193

Determine the order and degree of the following differential equation:

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

Exercise 6.1 | Q 8 | Page 193

Determine the order and degree of the following differential equation:

`[1 + ("dy"/"dx")^2]^(3/2) = 8 ("d"^2"y")/"dx"^2`

Exercise 6.1 | Q 9 | Page 193

Determine the order and degree of the following differential equation:

`(("d"^3"y")/"dx"^3)^(1/2) * ("dy"/"dx")^(1/3) = 20`

Exercise 6.1 | Q 10 | Page 193

Determine the order and degree of the following differential equation:

`"x" + ("d"^2"y")/"dx"^2 = sqrt(1 + (("d"^2"y")/"dx"^2)^2)`

Exercise 6.2 [Page 196]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 6 Differential Equations Exercise 6.2 [Page 196]

Exercise 6.2 | Q 1.01 | Page 196

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

x3 + y3 = 4ax

Exercise 6.2 | Q 1.02 | Page 196

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

Ax2 + By2 = 1

Exercise 6.2 | Q 1.03 | Page 196

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

y = A cos (log x) + B sin (log x)

Exercise 6.2 | Q 1.04 | Page 196

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

y2 = (x + c)3

Exercise 6.2 | Q 1.05 | Page 196

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

y = Ae5x + Be-5x 

Exercise 6.2 | Q 1.06 | Page 196

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

(y - a)2 = 4(x - b)

Exercise 6.2 | Q 1.07 | Page 196

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

y = a + `"a"/"x"`

Exercise 6.2 | Q 1.08 | Page 196

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

y = c1e2x + c2e5x 

Exercise 6.2 | Q 1.09 | Page 196

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

c1x3 + c2y2 = 5

Exercise 6.2 | Q 1.1 | Page 196

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

y = e-2x (A cos x + B sin x)

Exercise 6.2 | Q 2 | Page 196

Form the differential equation of the family of lines having intercepts a and b on the coordinate axes respectively.

Exercise 6.2 | Q 3 | Page 196

Find the differential equation all parabolas having a length of latus rectum 4a and axis is parallel to the axis.

Exercise 6.2 | Q 4 | Page 196

Find the differential equation of the ellipse whose major axis is twice its minor axis.

Exercise 6.2 | Q 5 | Page 196

Form the differential equation of family of lines parallel to the line 2x + 3y + 4 = 0.

Exercise 6.2 | Q 6 | Page 196

Find the differential equation of all circles having radius 9 and centre at point (h, k).

Exercise 6.2 | Q 7 | Page 196

Form the differential equation of all parabolas whose axis is the X-axis.

Exercise 6.3 [Pages 200 - 201]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 6 Differential Equations Exercise 6.3 [Pages 200 - 201]

Exercise 6.3 | Q 1.1 | Page 200

In the following example verify that the given expression is a solution of the corresponding differential equation:

xy = log y +c; `"dy"/"dx" = "y"^2/(1 - "xy")`

Exercise 6.3 | Q 1.2 | Page 200

In the following example verify that the given expression is a solution of the corresponding differential equation:

y = `(sin^-1 "x")^2 + "c"; (1 - "x"^2) ("d"^2"y")/"dx"^2 - "x" "dy"/"dx" = 2`

Exercise 6.3 | Q 1.3 | Page 200

In the following example verify that the given expression is a solution of the corresponding differential equation:

y = e-x + Ax + B; `"e"^"x" ("d"^2"y")/"dx"^2 = 1`

Exercise 6.3 | Q 1.4 | Page 200

In the following example verify that the given expression is a solution of the corresponding differential equation:

y = xm; `"x"^2 ("d"^2"y")/"dx"^2 - "mx" "dy"/"dx" + "my" = 0`

Exercise 6.3 | Q 1.5 | Page 200

In the following example verify that the given expression is a solution of the corresponding differential equation:

y = `"a" + "b"/"x"; "x" ("d"^2"y")/"dx"^2 + 2 "dy"/"dx" = 0`

Exercise 6.3 | Q 1.6 | Page 200

In the following example verify that the given expression is a solution of the corresponding differential equation:

y = `"e"^"ax"; "x" "dy"/"dx" = "y" log "y"`

Exercise 6.3 | Q 2.01 | Page 201

Solve the following differential equation:

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

Exercise 6.3 | Q 2.02 | Page 201

Solve the following differential equation:

`log  ("dy"/"dx") = 2"x" + 3"y"`

Exercise 6.3 | Q 2.03 | Page 201

Solve the following differential equation:

`"y" - "x" "dy"/"dx" = 0`

Exercise 6.3 | Q 2.04 | Page 201

Solve the following differential equation:

`"sec"^2 "x" * "tan y"  "dx" + "sec"^2 "y" * "tan x"  "dy" = 0` 

Exercise 6.3 | Q 2.05 | Page 201

Solve the following differential equation:

`cos "x" * cos "y"  "dy" - sin "x" * "sin y" "dx" = 0`

Exercise 6.3 | Q 2.06 | Page 201

Solve the following differential equation:

`"dy"/"dx" = - "k",` where k is a constant.

Exercise 6.3 | Q 2.07 | Page 201

Solve the following differential equation:

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

Exercise 6.3 | Q 2.08 | Page 201

Solve the following differential equation:

`"y"^3 - "dy"/"dx" = "x"^2 "dy"/"dx"`

Exercise 6.3 | Q 2.09 | Page 201

Solve the following differential equation:

`2"e"^("x + 2y") "dx" - 3"dy" = 0`

Exercise 6.3 | Q 2.1 | Page 201

Solve the following differential equation:

`"dy"/"dx" = "e"^("x + y") + "x"^2 "e"^"y"`

Exercise 6.3 | Q 3.1 | Page 201

For the following differential equation find the particular solution satisfying the given condition:

3ex tan y dx + (1 + ex) sec2 y dy = 0, when x = 0, y = π.

Exercise 6.3 | Q 3.2 | Page 201

For the following differential equation find the particular solution satisfying the given condition:

`("x" - "y"^2"x") "dx" - ("y" + "x"^2"y")"dy" = 0,` when x = 2, y = 0

Exercise 6.3 | Q 3.3 | Page 201

For the following differential equation find the particular solution satisfying the given condition:

`"y"(1 + log "x") "dx"/"dy" - "x" log "x" = 0, "y" = "e"^2,` when x = e

Exercise 6.3 | Q 3.4 | Page 201

For the following differential equation find the particular solution satisfying the given condition:

`("e"^"y" + 1) cos "x" + "e"^"y" sin "x" "dy"/"dx" = 0,  "when" "x" = pi/6,` y = 0

Exercise 6.3 | Q 3.5 | Page 201

For the following differential equation find the particular solution satisfying the given condition:

`("x" + 1) "dy"/"dx" - 1 = 2"e"^-"y" , "y" = 0`, when x = 1

Exercise 6.3 | Q 3.6 | Page 201

For the following differential equation find the particular solution satisfying the given condition:

`cos("dy"/"dx") = "a", "a" ∈ "R", "y"(0) = 2`

Exercise 6.3 | Q 4.1 | Page 201

Reduce the following differential equation to the variable separable form and hence solve:

`"dy"/"dx" = cos("x + y")`

Exercise 6.3 | Q 4.2 | Page 201

Reduce the following differential equation to the variable separable form and hence solve:

`("x - y")^2 "dy"/"dx" = "a"^2`

Exercise 6.3 | Q 4.3 | Page 201

Reduce the following differential equation to the variable separable form and hence solve:

`"x + y""dy"/"dx" = sec("x"^2 + "y"^2)`

Exercise 6.3 | Q 4.4 | Page 201

Reduce the following differential equation to the variable separable form and hence solve:

`cos^2 ("x - 2y") = 1 - 2 "dy"/"dx"`

Exercise 6.3 | Q 4.5 | Page 201

Reduce the following differential equation to the variable separable form and hence solve:

(2x - 2y + 3)dx - (x - y + 1)dy = 0, when x = 0, y = 1.

Exercise 6.4 [Page 203]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 6 Differential Equations Exercise 6.4 [Page 203]

Exercise 6.4 | Q 1 | Page 203

Solve the following differential equation:

`"x" sin ("y"/"x") "dy" = ["y" sin ("y"/"x") - "x"] "dx"`

Exercise 6.4 | Q 2 | Page 203

Solve the following differential equation:

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

Exercise 6.4 | Q 3 | Page 203

Solve the following differential equation:

`(1 + 2"e"^("x"/"y")) + 2"e"^("x"/"y")(1 - "x"/"y") "dy"/"dx" = 0`

Exercise 6.4 | Q 4 | Page 203

Solve the following differential equation:

y2 dx + (xy + x2)dy = 0

Exercise 6.4 | Q 5 | Page 203

Solve the following differential equation:

(x2 - y2)dx + 2xy dy = 0 

Exercise 6.4 | Q 6 | Page 203

Solve the following differential equation:

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

Exercise 6.4 | Q 7 | Page 203

Solve the following differential equation:

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

Exercise 6.4 | Q 8 | Page 203

Solve the following differential equation:

`(1 + "e"^("x"/"y"))"dx" + "e"^("x"/"y")(1 - "x"/"y")"dy" = 0`

Exercise 6.4 | Q 9 | Page 203

Solve the following differential equation:

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

Exercise 6.4 | Q 10 | Page 203

Solve the following differential equation:

`"xy" "dy"/"dx" = "x"^2 + "2y"^2, "y"(1) = 0`

Exercise 6.4 | Q 11 | Page 203

Solve the following differential equation:

x dx + 2y dx = 0, when x = 2, y = 1

Exercise 6.4 | Q 12 | Page 203

Solve the following differential equation:

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

Exercise 6.4 | Q 13 | Page 203

Solve the following differential equation:

(9x + 5y) dy + (15x + 11y)dx = 0

Exercise 6.4 | Q 14 | Page 203

Solve the following differential equation:

(x2 + 3xy + y2)dx - x2 dy = 0

Exercise 6.4 | Q 15 | Page 203

Solve the following differential equation:

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

Exercise 6.5 [Pages 206 - 207]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 6 Differential Equations Exercise 6.5 [Pages 206 - 207]

Exercise 6.5 | Q 1.01 | Page 206

Solve the following differential equation:

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

Exercise 6.5 | Q 1.02 | Page 206

Solve the following differential equation:

`cos^2 "x" * "dy"/"dx" + "y" = tan "x"`

Exercise 6.5 | Q 1.03 | Page 206

Solve the following differential equation:

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

Exercise 6.5 | Q 1.04 | Page 206

Solve the following differential equation:

`"dy"/"dx" + "y" * sec "x" = tan "x"`

Exercise 6.5 | Q 1.05 | Page 206

Solve the following differential equation:

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

Exercise 6.5 | Q 1.06 | Page 206

Solve the following differential equation:

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

Exercise 6.5 | Q 1.07 | Page 206

Solve the following differential equation:

`("x + a")"dy"/"dx" - 3"y" = ("x + a")^5`

Exercise 6.5 | Q 1.08 | Page 206

Solve the following differential equation:

dr + (2r cot θ + sin 2θ) dθ = 0

Exercise 6.5 | Q 1.09 | Page 206

Solve the following differential equation:

y dx + (x - y2) dy = 0

Exercise 6.5 | Q 1.1 | Page 207

Solve the following differential equation:

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

Exercise 6.5 | Q 1.11 | Page 207

Solve the following differential equation:

`(1 + "x"^2) "dy"/"dx" + "y" = "e"^(tan^-1 "x")`

Exercise 6.5 | Q 2 | Page 207

Find the equation of the curve which passes through the origin and has the slope x + 3y - 1 at any point (x, y) on it.

Exercise 6.5 | Q 3 | Page 207

Find the equation of the curve passing through the point `(3/sqrt2, sqrt2)` having a slope of the tangent to the curve at any point (x, y) is -`"4x"/"9y"`.

Exercise 6.5 | Q 4 | Page 207

The curve passes through the point (0, 2). The sum of the coordinates of any point on the curve exceeds the slope of the tangent to the curve at any point by 5. Find the equation of the curve.

Exercise 6.5 | Q 5 | Page 207

If the slope of the tangent to the curve at each of its point is equal to the sum of abscissa and the product of the abscissa and ordinate of the point. Also, the curve passes through the point (0, 1). Find the equation of the curve.

Exercise 6.6 [Page 213]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 6 Differential Equations Exercise 6.6 [Page 213]

Exercise 6.6 | Q 1 | Page 213

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 6.6 | Q 2 | Page 213

If the population of a country doubles in 60 years; in how many years will it be triple (treble) under the assumption that the rate of increase is proportional to the number of inhabitants?
(Given log = 20.6912, log 3 = 1.0986)

Exercise 6.6 | Q 3 | Page 213

If a body cools from 80°C to 50°C at room temperature of 25°C in 30 minutes, find the temperature of the body after 1 hour.

Exercise 6.6 | Q 4 | Page 213

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 `2 1/2` hours.
[Take `sqrt2 = 1.414`]

Exercise 6.6 | Q 5 | Page 213

The rate of disintegration of a radioactive element at any time t is proportional to its mass at that time. Find the time during which the original mass of 1.5 gm will disintegrate into its mass of 0.5 gm.

Exercise 6.6 | Q 6 | Page 213

The rate of decay of certain substances is directly proportional to the amount present at that instant. Initially, there is 25 gm of certain substance and two hours later it is found that 9 gm are left. Find the amount left after one more hour.

Exercise 6.6 | Q 7 | Page 213

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 6.6 | Q 8 | Page 213

A body cools according to Newton’s law from 100° C to 60° C in 20 minutes. The temperature of the surrounding being 20° C. How long will it take to cool down to 30° C?

Exercise 6.6 | Q 9 | Page 213

A right circular cone has height 9 cm and radius of the base 5 cm. It is inverted and water is poured into it. If at any instant the water level rises at the rate of `(pi/"A")`cm/sec, where A is the area of the water surface A at that instant, show that the vessel will be full in 75 seconds.

Exercise 6.6 | Q 10 | Page 213

Assume that a spherical raindrop evaporates at a rate proportional to its surface area. If its radius originally is 3 mm and 1 hour later has been reduced to 2 mm, find an expression for the radius of the raindrop at any time t.

Exercise 6.6 | Q 11 | Page 213

The rate of growth of the population of a city at any time t is proportional to the size of the population. For a certain city, it is found that the constant of proportionality is 0.04. Find the population of the city after 25 years, if the initial population is 10,000. [Take e = 2.7182]

Exercise 6.6 | Q 12 | Page 213

Radium decomposes at the rate proportional to the amount present at any time. If p percent of the amount disappears in one year, what percent of the amount of radium will be left after 2 years?

Miscellaneous exercise 1 [Pages 214 - 216]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 6 Differential Equations Miscellaneous exercise 1 [Pages 214 - 216]

Miscellaneous exercise 1 | Q 1.01 | Page 214

Choose the correct option from the given alternatives:

The order and degree of the differential equation `sqrt(1 + ("dy"/"dx")^2) = (("d"^2"y")/"dx"^2)^(3/2)` are respectively.

  • 2, 1

  • 1, 2

  • 3, 2

  • 2, 3

Miscellaneous exercise 1 | Q 1.02 | Page 214

Choose the correct option from the given alternatives:

The differential equation of y = `"c"^2 + "c"/"x"` is

  • `"x"^4 ("dy"/"dx")^2 - "x" "dy"/"dx" = "y"`

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

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

  • `("d"^2"y")/"dx"^2 + "dy"/"dx" - "y" = 0`

Miscellaneous exercise 1 | Q 1.03 | Page 215

Choose the correct option from the given alternatives:

x2 + y2 = a2 is a solution of

  • `("d"^2"y")/"dx"^2 + "dy"/"dx" - "y" = 0`

  • y = x`sqrt(1 + ("dy"/"dx")^2) + "a"^2 "y"`

  • y = x`"dy"/"dx" + "a" sqrt(1 + ("dy"/"dx")^2)`

  • `("d"^2"y")/"dx"^2 = ("x + 1")"dy"/"dx"`

Miscellaneous exercise 1 | Q 1.04 | Page 215

Choose the correct option from the given alternatives:

The differential equation of all circles having their centres on the line y = 5 and touching the X-axis is

  • `"y"^2 (1 + "dy"/"dx") = 25`

  • `("y - 5")^2 [1 + ("dy"/"dx")^2] = 25`

  • `("y - 5")^2 + [1 + ("dy"/"dx")^2] = 25`

  • `("y - 5")^2 [1 - ("dy"/"dx")^2] = 25`

Miscellaneous exercise 1 | Q 1.05 | Page 215

Choose the correct option from the given alternatives:

The differential equation `"y" "dy"/"dx" + "x" = 0` represents family of

  • circles

  • parabolas

  • ellipses

  • hyperbolas

Miscellaneous exercise 1 | Q 1.06 | Page 215

Choose the correct option from the given alternatives:

The solution of `1/"x" * "dy"/"dx" = tan^-1 "x"` is

  • `("x"^2 tan^-1 "x")/2 + "c" = 0`

  • x tan-1 x + c = 0

  • x - tan-1 x = c 

  • y = `("x"^2 tan^-1 "x")/2 - 1/2 ("x" - tan^-1 "x") + "c"`

Miscellaneous exercise 1 | Q 1.07 | Page 215

Choose the correct option from the given alternatives:

The solution of `("x + y")^2 "dy"/"dx" = 1` is

  • x = tan-1 (x + y) + c

  • y tan-1 `("x"/"y") = "c"`

  • y = tan-1 (x + y) + c

  • y + tan-1 (x + y) + c

Miscellaneous exercise 1 | Q 1.08 | Page 215

Choose the correct option from the given alternatives:

The solution of `"dy"/"dx" = ("y" + sqrt("x"^2 - "y"^2))/"x"` is

  • `sin^-1 ("y"/"x") = 2 log |"x"| + "c"`

  • `sin^-1 ("y"/"x") =  log |"x"| + "c"`

  • `sin ("y"/"x") = log |"x"| + "c"`

  • `sin ("y"/"x") = 2 log |"x"| + "c"`

Miscellaneous exercise 1 | Q 1.09 | Page 215

Choose the correct option from the given alternatives:

The solution of `"dy"/"dx" + "y" = cos "x" - sin "x"`

  • yex = cos x + c

  • yex + ex  cos x = c

  • yex = ex cos x + c

  • y2ex = ex cos x + c

Miscellaneous exercise 1 | Q 1.1 | Page 216

Choose the correct option from the given alternatives:

The integrating factor of linear differential equation `"x" "dy"/"dx" + "2y" = "x"^2 log "x"` is

  • `1/"x"`

  • k

  • `1/"n"^2`

  • x2 

Miscellaneous exercise 1 | Q 1.11 | Page 216

Choose the correct option from the given alternatives:

The solution of the differential equation `"dy"/"dx" = sec "x" - "y" tan "x"`

  • y sec x + tan x = c

  • y sec x = tan x + c

  • sec x + y tan x = c

  • sec x = y tan x + c

Miscellaneous exercise 1 | Q 1.12 | Page 216

Choose the correct option from the given alternatives:

The particular solution of `"dy"/"dx" = "x""e"^("y - x")`, when x = y = 0 is

  • `"e"^("x - y") = "x + 1"`

  • `"e"^("x + y")` = x + 1

  • `"e"^"x" + "e"^"y"` = x + 1

  • `"e"^("y - x")` = x - 1

Miscellaneous exercise 1 | Q 1.13 | Page 216

Choose the correct option from the given alternatives:

`"x"^2/"a"^2 - "y"^2/"b"^2 = 1` is a solution of

  • `("d"^2"y")/"dx"^2 + "yx" + ("dy"/"dx")^2 = 0`

  • `"xy"*("d"^2"y")/"dx"^2 + "x"("dy"/"dx")^2 - "y" "dy"/"dx" = 0`

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

  • `"xy" "dy"/"dx" + "y" ("d"^2"y")/"dx"^2 = 0`

Miscellaneous exercise 1 | Q 1.14 | Page 216

Choose the correct option from the given alternatives:

The decay rate of certain substances is directly proportional to the amount present at that instant. Initially there are 27 grams of substance and 3 hours later it is found that 8 grams left. The amount left after one more hour is

  • `5 2/3` grams

  • `5 1/3` grams

  • 5.1 grams

  • 5 grams

Miscellaneous exercise 1 | Q 1.15 | Page 216

Choose the correct option from the given alternatives:

If the surrounding air is kept at 20° C and a body cools from 80° C to 70° C in 5 minutes, the temperature of the body after 15 minutes will be

  • 51.7° C

  • 54.7° C

  • 52.7° C

  • 50.7° C

Miscellaneous exercise 1 | Q 1.15 | Page 216

Choose the correct option from the given alternatives:

If the surrounding air is kept at 20° C and a body cools from 80° C to 70° C in 5 minutes, the temperature of the body after 15 minutes will be

  • 51.7° C

  • 54.7° C

  • 52.7° C

  • 50.7° C

Miscellaneous exercise 2 [Pages 216 - 218]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 6 Differential Equations Miscellaneous exercise 2 [Pages 216 - 218]

Miscellaneous exercise 2 | Q 1.1 | Page 216

Determine the order and degree of the following differential equation:

`("d"^2"y")/"dx"^2 + 5 "dy"/"dx" + "y" = "x"^3`

Miscellaneous exercise 2 | Q 1.2 | Page 216

Determine the order and degree of the following differential equation:

`(("d"^3"y")/"dx"^3)^2 = root(5)(1 + "dy"/"dx")`

Miscellaneous exercise 2 | Q 1.3 | Page 216

Determine the order and degree of the following differential equation:

`root(3)(1 +("dy"/"dx")^2) = ("d"^2"y")/"dx"^2`

Miscellaneous exercise 2 | Q 1.4 | Page 216

Determine the order and degree of the following differential equation:

`"dy"/"dx" = 3"y" + root(4)(1 + 5 ("dy"/"dx")^2)`

Miscellaneous exercise 2 | Q 1.5 | Page 216

Determine the order and degree of the following differential equation:

`("d"^4"y")/"dx"^4 + sin ("dy"/"dx") = 0`

Miscellaneous exercise 2 | Q 2.1 | Page 217

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

`"x"^2 + "y"^2 = "r"^2; "x" "dy"/"dx" + "r" sqrt(1 + ("dy"/"dx")^2) = "y"`

Miscellaneous exercise 2 | Q 2.2 | Page 217

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

`"y" = "e"^"ax" sin "bx"; ("d"^2"y")/"dx"^2 - 2"a" "dy"/"dx" + ("a"^2 + "b"^2)"y" = 0`

Miscellaneous exercise 2 | Q 2.3 | Page 217

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

`"y" = 3 "cos" (log "x") + 4 sin (log "x"); "x"^2 ("d"^2"y")/"dx"^2 + "x" "dy"/"dx" + "y" = 0`

Miscellaneous exercise 2 | Q 2.4 | Page 217

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

`"xy" = "ae"^"x" + "be"^-"x" + "x"^2; "x" ("d"^2"y")/"dx"^2 + 2 "dy"/"dx" + "x"^2 = "xy" + 2`

Miscellaneous exercise 2 | Q 2.5 | Page 217

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

`"x"^2 = "2y"^2 log "y",  "x"^2 + "y"^2 = "xy" "dx"/"dy"`

Miscellaneous exercise 2 | Q 3.1 | Page 217

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

`"y"^2 = "a"("b - x")("b + x")`

Miscellaneous exercise 2 | Q 3.2 | Page 217

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

y = a sin (x + b)

Miscellaneous exercise 2 | Q 3.3 | Page 217

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

(y - a)2 = b(x + 4)

Miscellaneous exercise 2 | Q 3.4 | Page 217

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

y = `sqrt("a" cos (log "x") + "b" sin (log "x"))`

Miscellaneous exercise 2 | Q 3.5 | Page 217

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

y = `"Ae"^(3"x" + 1) + "Be"^(- 3"x" + 1)`

Miscellaneous exercise 2 | Q 4.1 | Page 217

Form the differential equation of all circles which pass through the origin and whose centers lie on X-axis.

Miscellaneous exercise 2 | Q 4.2 | Page 217

Form the differential equation of all parabolas which have 4b as latus rectum and whose axis is parallel to the Y-axis.

Miscellaneous exercise 2 | Q 4.3 | Page 217

Find the differential equation of the ellipse whose major axis is twice its minor axis.

Miscellaneous exercise 2 | Q 4.4 | Page 217

Form the differential equation of all the lines which are normal to the line 3x + 2y + 7 = 0.

Miscellaneous exercise 2 | Q 4.5 | Page 217

Form the differential equation of the hyperbola whose length of transverse and conjugate axes are half of that of the given hyperbola `"x"^2/16 - "y"^2/36 = "k"`.

Miscellaneous exercise 2 | Q 5.1 | Page 217

Solve the following differential equation:

`log  ("dy"/"dx") = 2"x" + 3"y"`

Miscellaneous exercise 2 | Q 5.2 | Page 217

Solve the following differential equation:

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

Miscellaneous exercise 2 | Q 5.3 | Page 217

Solve the following differential equation:

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

Miscellaneous exercise 2 | Q 5.4 | Page 217

Solve the following differential equation:

x dy = (x + y + 1) dx

Miscellaneous exercise 2 | Q 5.5 | Page 217

Solve the following differential equation:

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

Miscellaneous exercise 2 | Q 5.6 | Page 217

Solve the following differential equation:

y log y = (log y2 - x) `"dy"/"dx"`

Miscellaneous exercise 2 | Q 5.7 | Page 217

Solve the following differential equation:

`"dx"/"dy" + "8x" = 5"e"^(- 3"y")`

Miscellaneous exercise 2 | Q 6.1 | Page 218

Find the particular solution of the following differential equation:

y(1 + log x) = (log xx) `"dy"/"dx"`, when y(e) = e2

Miscellaneous exercise 2 | Q 6.2 | Page 218

Find the particular solution of the following differential equation:

`("x + 2y"^2) "dy"/"dx" = "y",` when x = 2, y = 1

Miscellaneous exercise 2 | Q 6.3 | Page 218

Find the particular solution of the following differential equation:

`"dy"/"dx" - 3"y" cot "x" = sin "2x"`, when `"y"(pi/2) = 2`

Miscellaneous exercise 2 | Q 6.4 | Page 218

Find the particular solution of the following differential equation:

(x + y)dy + (x - y)dx = 0; when x = 1 = y

Miscellaneous exercise 2 | Q 6.5 | Page 218

Find the particular solution of the following differential equation:

`(1 + 2"e"^("x"//"y"))"dx" + 2"e"^("x"//"y")(1 - "x"/"y")"dy" = 0` when y(0) = 1

Miscellaneous exercise 2 | Q 7 | Page 218

Show that the general solution of differential equation `"dy"/"dx" + ("y"^2 + "y" + 1)/("x"^2 + "x" + 1) = 0` is given by (x + y + 1) = (1 - x - y - 2xy).

Miscellaneous exercise 2 | Q 8 | Page 218

The normal lines to a given curve at each point (x, y) on the curve pass through (2, 0). The curve passes through (2, 3). Find the equation of the curve.

Miscellaneous exercise 2 | Q 9 | Page 218

The volume of a spherical balloon 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.

Miscellaneous exercise 2 | Q 10 | Page 218

A person’s assets start reducing in such a way that the rate of reduction of assets is proportional to the square root of the assets existing at that moment. If the assets at the beginning ax ‘ 10 lakhs and they dwindle down to ‘ 10,000 after 2 years, show that the person will be bankrupt in `2 2/9` years from the start.

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Chapter 6: Differential Equations

Exercise 6.1Exercise 6.2Exercise 6.3Exercise 6.4Exercise 6.5Exercise 6.6Miscellaneous exercise 1Miscellaneous exercise 2
Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 6 - Differential Equations - Shaalaa.com

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 6 - Differential Equations

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 6 (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 Maharashtra State Board Mathematics and Statistics 2 (Arts and Science) 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 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 6 Differential Equations are Differential Equations, Order and Degree of a Differential Equation, Formation of Differential Equations, Homogeneous Differential Equations, Linear Differential Equations, Application of Differential Equations.

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