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

Determine the order and degree of the following differential equation:

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

Determine the order and degree of the following differential equation:

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

Determine the order and degree of the following differential equation:

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

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)`

Determine the order and degree of the following differential equation:

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

Determine the order and degree of the following differential equation:

(y''')² + 3y'' + 3xy' + 5y = 0

Determine the order and degree of the following differential equation:

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

Determine the order and degree of the following differential equation:

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

Determine the order and degree of the following differential equation:

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

Determine the order and degree of the following differential equation:

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

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

x³ + y³ = 4ax

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

Ax² + By² = 1

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

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

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

y² = (x + c)³

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

y = Ae^5x + Be^-5x 

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

(y - a)² = 4(x - b)

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

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

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

y = c₁e^2x + c₂e^5x 

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

c₁x³ + c₂y² = 5

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

y = e^−2x (A cos x + B sin x)

Form the differential equation of family of lines having intercepts a and b on the co-ordinate ares respectively.

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

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

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

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

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

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")`

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`

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`

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

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

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`

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"`

Solve the following differential equation:

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

Solve the following differential equation:

`log  ("dy"/"dx") = 2"x" + 3"y"`

Solve the following differential equation:

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

Solve the following differential equation:

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

Solve the following differential equation:

cos x . cos y dy − sin x . sin y dx = 0

Solve the following differential equation:

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

Solve the following differential equation:

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

Solve the following differential equation:

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

Solve the following differential equation:

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

Solve the following differential equation:

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

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

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

For the differential equation, find the particular solution (x – y²x) dx – (y + x²y) dy = 0 when x = 2, y = 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

Solve the following differential equation:

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

Solve the following differential equation:

(x² + y²)dx - 2xy dy = 0

Solve the following differential equation:

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

Solve the following differential equation:

y² dx + (xy + x²)dy = 0

Solve the following differential equation:

(x² – y²)dx + 2xy dy = 0

Solve the following differential equation:

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

Solve the following differential equation:

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

Solve the following differential equation:

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

Solve the following differential equation:

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

Solve the following differential equation:

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

Solve the following differential equation:

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

Solve the following differential equation:

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

Solve the following differential equation:

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

Solve the following differential equation:

(x² + 3xy + y2)dx - x² dy = 0

Solve the following differential equation:

(x² + y²)dx - 2xy dy = 0

Solve the following differential equation:

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

Solve the following differential equation:

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

Solve the following differential equation:

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

Solve the following differential equation:

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

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:

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

Solve the following differential equation dr + (2r cot θ + sin 2θ) dθ = 0.

Solve the following differential equation:

y dx + (x - y²) dy = 0

Solve the following differential equation:

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

Solve the following differential equation:

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

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.

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"`.

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.

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.

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.

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 2 = 0.6912, log 3 = 1.0986)

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.

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`]

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.

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.

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.

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?

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.

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.

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]

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?

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.

Choose the correct option from the given alternatives:

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

Choose the correct option from the given alternatives:

x² + y² = a² is a solution of

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

Choose the correct option from the given alternatives:

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

Choose the correct option from the given alternatives:

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

Choose the correct option from the given alternatives:

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

Choose the correct option from the given alternatives:

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

Choose the correct option from the given alternatives:

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

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

Choose the correct option from the given alternatives:

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

The particular solution of `dy/dx = xe^(y - x)`, when x = y = 0 is ______.

Choose the correct option from the given alternatives:

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

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

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

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

Determine the order and degree of the following differential equation:

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

Determine the order and degree of the following differential equation:

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

Determine the order and degree of the following differential equation:

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

Determine the order and degree of the following differential equation:

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

Determine the order and degree of the following differential equation:

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

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"`

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`

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`

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`

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"`

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

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

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

y = a sin (x + b)

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

(y - a)² = b(x + 4)

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

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

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

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

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

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

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

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

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"`.

Solve the following differential equation:

`log  ("dy"/"dx") = 2"x" + 3"y"`

Solve the following differential equation:

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

Solve the following differential equation:

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

Solve the following differential equation:

x dy = (x + y + 1) dx

Solve the following differential equation:

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

Solve the following differential equation:

y log y = (log y² - x) `"dy"/"dx"`

Solve the following differential equation:

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

Find the particular solution of the following differential equation:

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

Find the particular solution of the following differential equation:

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

Find the particular solution of the following differential equation:

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

Find the particular solution of the following differential equation:

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

Find the particular solution of the following differential equation:

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

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).

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.

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.

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.