Samacheer Kalvi solutions for Mathematics - Volume 1 and 2 [English] Class 12 TN Board chapter 10 - Ordinary Differential Equations [Latest edition]

For the following equations, determine its order, degree (if exists)

`("d"y)/("d"x) + xy` = cot x

For the following equations, determine its order, degree (if exists)

`(("d"^3y)/("d"x^3))^(2/3) - 3 ("d"^2y)/("d"x^2) + 5("d"y)/("d"x) + 4` = 0

For the following equations, determine its order, degree (if exists)

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

For the following equations, determine its order, degree (if exists)

`sqrt(("d"y)/("d"x)) - 4 ("d"y)/("d"x) - 7x` = 0

For the following equations, determine its order, degree (if exists)

`y(("d"y)/("d"x)) = x/((("d"y)/("d"x)) + (("d"y)/("d"x))^3`

For the following equations, determine its order, degree (if exists)

`x^2 ("d"^2y)/("d"x^2) + [1 + (("d"y)/("d"x))^2]^(1/2)` = 0

For the following equations, determine its order, degree (if exists)

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

For the following equations, determine its order, degree (if exists)

`("d"^2y)/("d"x^2) = xy + cos(("d"y)/("d"x))`

For the following equations, determine its order, degree (if exists)

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

For the following equations, determine its order, degree (if exists)

x = `"e"^(xy((dy)/(dx))`

Express the following physical statements in the form of differential equation.

Radium decays at a rate proportional to the amount Q present

Express the following physical statements in the form of differential equation.

The population P of a city increases at a rate proportional to the product of population and to the difference between 5,00,000 and the population

Express the following physical statements in the form of differential equation.

For a certain substance, the rate of change of vapor pressure P with respect to temperature T is proportional to the vapor pressure and inversely proportional to the square of the temperature

Express the following physical statements in the form of differential equation.

A saving amount pays 8% interest per year compound continuously. In addition, the income from another investment is credited to the amount continuously at the rate of ₹ 400 per year.

Assume that a spherical rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop

Find the differential equation of the family of all non-vertical lines in a plane

Find the differential equation of the family of all non-horizontal lines in a plane 

Form the differential equation of all straight lines touching the circle x² + y² = r²

Find the differential equation of the family of circles passing through the origin and having their centres on the x-axis

Find the differential equation of the family of all the parabolas with latus rectum 4a and whose axes are parallel to the x-axis

Find the differential equation of the family of parabolas with vertex at (0, –1) and having axis along the y-axis

Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at the origin

Find the differential equation corresponding to the family of curves represented by the equation y = Ae^8x + Be ^–8x, where A and B are arbitrary constants

Find the differential equation of the curve represented by xy = ae^x + be^–x + x²

Show the following expressions is a solution of the corresponding given differential equation.

y = 2x² ; xy’ = 2y

Show the following expressions is a solution of the corresponding given differential equation.

y = ae^x + be^–x ; y'' – y = 0

Find the value of m so that the function y = e^mx solution of the given differential equation.

y’ + 2y = 0

Find the value of m so that the function y = e^mx solution of the given differential equation.

y” – 5y’ + 6y = 0

The Slope of the tangent to the curve at any point is the reciprocal of four times the ordinate at that point. The curve passes through (2, 5). Find the equation of the curve

Show that y = e^–x + mx + n is a solution of the differential equation `"e"^x(("d"^2y)/("d"x^2)) - 1` = 0

Show that y = `"a"x + "b"/x ≠ 0` is a solution of the differential equation x²yⁿ + xy’ – y = 0

Show that y = ae^–3x + b, where a and b are arbitrary constants, is a solution of the differential equation `("d"^2y)/("d"x^2)  + 3("d"y)/("d"x)` = 0

Show that the differential equation representing the family of curves y² = `2"a"(x + "a"^(2/3))`, where a is a postive parameter, is `(y^2 - 2xy ("d"y)/("d"x))^3 = 8(y ("d"y)/("d"x))^5`

Show that y = a cos bx is a solution of the! differential equation `("d"^2y)/("d"x^2) + "b"^2y` = 0

If F is the constant force generated by the motor of an automobile of mass M, its velocity V is given by `"M""dv"/"dt"` = F – kV, where k is a constant. Express V in terms of t given that V = 0 when t = 0

The velocity v, of a parachute falling vertically satisfies the equation `"v" (dv)/(dx) = "g"(1 - v^2/k^2)` where g and k are constants. If v and are both initially zero, find v in terms of x

Find the equation of the curve whose slope is `(y - 1)/(x^2 + x)` and which passes through the point (1, 0)

Solve the following differential equation:

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

Solve the following differential equation:

`y"d"x + (1 + x^2)tan^-1x  "d"y`= 0

Solve the following differential equation:

`sin  ("d"y)/("d"x)` = a, y(0) = 1

Solve the following differential equation:

`("d"y)/("d"x) = "e"^(x + y) - x^3"e"^y`

Solve the following differential equation:

(e^y + 1)cos x dx + e^y sin x dy = 0

Solve the following differential equation:

`(ydx - xdy) cot (x/y)` = ny² dx

Solve the following differential equation:

`("d"y)/("d"x) - xsqrt(25 - x^2)` = 0

Solve the following differential equation:

x cos y dy = e^x(x log x + 1) dx

Solve the following differential equation:

`tan y ("d"y)/("d"x) = cos(x + y) + cos(x - y)`

Solve the following differential equation:

`("d"y)/("d"x) = tan^2(x + y)`

Solve the following differential equation:

`[x + y cos(y/x)] "d"x = x cos(y/x) "d"y`

Solve the following differential equation:

`(x^3 + y^3)"d"y - x^2 y"d"x` = 0

Solve the following differential equation:

`y"e"^(x/y) "d"x = (x"e"^(x/y) + y) "d"y`

Solve the following differential equation:

`2xy"d"x + (x^2 + 2y^2)"d"y` = 0

Solve the following differential equation:

`(y^2 - 2xy) "d"x = (x^2 - 2xy) "d"y`

Solve the following differential equation:

`x ("d"y)/("d"x) = y - xcos^2(y/x)`

Solve the following differential equation:

`(1 + 3"e"^(y/x))"d"y + 3"e"^(y/x)(1 - y/x)"d"x` = 0, given that y = 0 when x = 1

Solve the following differential equation:

(x² + y²) dy = xy dx. It is given that y (1) = y(x₀) = e. Find the value of x₀

Solve the following Linear differential equation:

`cos x  ("d"y)/("d"x) + y sin x ` = 1

Solve the following Linear differential equation:

`(1 - x^2) ("d"y)/("d"x) - xy` = 1

Solve the following Linear differential equation:

`("d"y)/("d"x) + y/x = sin x`

Solve the following Linear differential equation:

`(x^2 + 1) ("d"y)/("d"x) + 2xy = sqrt(x^2 + 4)`

Solve the following Linear differential equation:

(2x – 10y³)dy + y dx = 0

Solve the following Linear differential equation:

`x sin x ("d"y)/("d"x) + (x cos x + sin x)y = sinx`

Solve the following Linear differential equation:

`(y - "e"^(sin^-1)x) ("d"x)/("d"y) + sqrt(1 - x^2)` = 0

Solve the following Linear differential equation:

`("d"y)/("d"x) + y/((1 - x)sqrt(x)) = 1 - sqrt(x)`

Solve the following Linear differential equation:

`(1 + x + xy^2) ("d"y)/("d"x) + (y + y^3)` = 0

Solve the following Linear differential equation:

`("d"y)/("d"x) + y/(xlogx) = (sin2x)/logx`

Solve the following Linear differential equation:

`(x + "a") ("d"y)/("d"x) - 2y = (x + "a")^4`

Solve the following Linear differential equation:

`("d"y)/("d"x) = (sin^2x)/(1 + x^3) - (3x^2)/(1 + x^3) y`

Solve the following Linear differential equation:

`x ("d"y)/("d"x) + y = x log x`

Solve the following Linear differential equation:

`x ("d"y)/("d"x) + 2y - x^2 log x` = 0

Solve the following Linear differential equation:

`("d"y)/("d"x) + (3y)/x = 1/x^2`, given that y = 2 when x = 1

The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given that the number triples in 5 hours, find how many bacteria will be present after 10 hours?

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 3,00,000 to 4,00,000

The equation of electromotive force for an electric circuit containing resistance and self-inductance is E = `"Ri"  + "L" "di"/"dt"`, where E is the electromotive force is given to the circuit, R the resistance and L, the coefficient of induction. Find the current i at time t when E = 0

The engine of a motor boat moving at 10 m/s is shut off. Given that the retardation at any subsequent time (after shutting off the engine) equal to the velocity at that time. Find the velocity after 2 seconds of switching off the engine

Suppose a person deposits ₹ 10,000 in a bank account at the rate of 5% per annum compounded continuously. How much money will be in his bank account 18 months later?

Assume that the rate at which radioactive nuclei decay is proportional to the number of such nuclei that are present in a given sample. In a certain sample, 10% of the original number of radioactive nuclei have undergone disintegration in a period of 100 years. What percentage of the original radioactive nuclei will remain after 1000 years?

Water at temperature 100°C cools in 10 minutes to 80°C at a room temperature of 25°C. Find the temperature of the water after 20 minutes

Water at temperature 100°C cools in 10 minutes to 80°C at a room temperature of 25°C. Find the time when the temperature is 40°C `[log_"e"  11/15 = - 0.3101; log_"e" 5 = 1.6094]`

At 10.00 A.M. a woman took a cup of hot instant coffee from her microwave oven and placed it on a nearby kitchen counter to cool. At this instant, the temperature of the coffee was 180°F and 10 minutes later it was 160°F. Assume that the constant temperature of the kitchen was 70°F. What was the temperature of the coffee at 10.15 AM? `|log  9/100 = - 0.6061|`

At 10.00 A.M. a woman took a cup of hot instant coffee from her microwave oven and placed it on a nearby kitchen counter to cool. At this instant, the temperature of the coffee was 180°F and 10 minutes later it was 160°F. Assume that the constant temperature of the kitchen was 70°F. The woman likes to drink coffee when its temperature is between130°F and 140°F. between what times should she have drunk the coffee? `|log  6/11 =  - 0.2006|`

A pot of boiling water at 100°C is removed from a stove at time t = 0 and left to cool in the kitchen. After 5 minutes, the water temperature has decreased to 80° C and another 5 minutes later it has dropped to 65°C. Determine the temperature of the kitchen

A tank initially contains 50 litres of pure water. Starting at time t = 0 a brine containing 2 grams of dissolved salt per litre flows into the tank at the rate of 3 litres per minute. The mixture is kept uniform by stirring and the well-stirred mixture simultaneously flows out of the tank at the same rate. Find the amount of salt present in the tank at any time t > 0

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The order and degree of the differential equation `("d"^2y)/("d"x^2) + (("d"y)/("d"x))^(1/3) + x^(1/4)` = 0 are respectively

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The differential equation representing the family of curves y = A cos (x + B), where A and B are parameters, is

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The order and degree of the differential equation `sqrt(sin  ) ("d"x + "d"y) = sqrt(cos  x)  ("d"x - "d"y)` is

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The order of the differential equation of all circles with centre at (h, k) and radius 'a' is

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The differential equation of the family of curves y = Ae^x + Be^-x, where A and B are arbitrary constants is

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The general solution of the differential equation `("d"y)/("d"x) = y/x` is

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The solution of the differential equation `2x ("d"y)/("d"x) - y = 3` represents

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The solution of `("d"y)/("d"x) + "p"(x)y = 0` is

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The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/lambda` is

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The Integrating factor of the differential equation `("d"y)/("d"x) + "P"(x)y = "Q"(x)` is x, then p(x)

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The degree of the differential equation `y(x) = 1 +  ("d"y)/("d"x) + 1/(1.2) (("d"y)/("d"x))^2 + 1/(1*2*3) (("d"y)/("d"x))^3 + ....` is

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If p and q are the order and degree of the differential equation `y("d"y)/("d"x) + x^3 (("d"^2y)/("d"x^2)) + xy = cos x`, when

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The solution of the differential equation `("d")/("d"x) + 1/sqrt(1 - x^2) = 0` is

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The solution of the differential equation `("d"y)/("d"x) = 2xy` is

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The general solution of the differential equation `log(("d"y)/("d"x)) = x + y` is

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The solution of `("d"y)/("d"x) = 2^(y - x)` is

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The solution of the differential equation `("d"y)/("d"x) = y/x + (∅(y/x))/(∅(y/x))` is

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If sin x is the integrating factor of the linear differential equation `("d"y)/("d"x) + "P"y = "Q"`, then P is

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The number of arbitrary constants in the general solutions of order n and n +1are respectively

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The number of arbitrary constants in the particular solution of a differential equation of third order is

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Integrating factor of the differential equation `("d"y)/("d"x) = (x + y + 1)/(x + 1)` is

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The population P in any year t is such that the rate of increase in the population is proportional to the population. Then

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P is the amount of certain substance left in after time t. If the rate of evaporation of the substance is proportional to the amount remaining, then

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If the solution of the differential equation `("d"y)/("d"x) = ("a"x + 3)/(2y + f)` represents a circle, then the value of a is

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The slope at any point of a curve y = f(x) is given by `("d"y)/("d"x) - 3x^2` and it passes through (-1, 1). Then the equation of the curve is