NCERT solutions for Mathematics Part 1 and 2 [English] Class 12 chapter 9 - Differential Equations [Latest edition]

Determine the order and degree (if defined) of the differential equation:

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

Determine the order and degree (if defined) of the differential equation:

y' + 5y = 0

Determine the order and degree (if defined) of the differential equation:

`((ds)/(dt))^4 + 3s  (d^2s)/(dt^2) = 0`

Determine the order and degree (if defined) of the differential equation:

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

Determine the order and degree (if defined) of the differential equation:

`(d^2y)/(dx^2)` = cos 3x + sin 3x

Determine the order and degree (if defined) of the differential equation:

( y′′′) + (y″)³ + (y′)⁴ + y⁵ = 0

Determine the order and degree (if defined) of the differential equation:

y′′′ + 2y″ + y′ = 0

Determine the order and degree (if defined) of the differential equation:

y′ + y = e^x

Determine the order and degree (if defined) of the differential equation:

y″ + (y′)² + 2y = 0

Determine the order and degree (if defined) of the differential equation:

y″ + 2y′ + sin y = 0

The degree of the differential equation `((d^2y)/(dx^2))^3 + ((dy)/(dx))^2 + sin ((dy)/(dx)) + 1 = 0` is ______.

The order of the differential equation `2x^2 (d^2y)/(dx^2) - 3 (dy)/(dx) + y = 0` is ______.

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:

y = e^x + 1  :  y″ – y′ = 0

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:

y = x² + 2x + C  :  y′ – 2x – 2 = 0

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:

y = cos x + C : y′ + sin x = 0

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:

`y sqrt(1 + x^2) : y' = (xy)/(1+x^2)`

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:

y = Ax : xy′ = y (x ≠ 0)

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:

y = x sin x : xy' = `y + x  sqrt (x^2 - y^2)`  (x ≠ 0 and x > y or x < -y)

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:

xy = log y + C :  `y' = (y^2)/(1 - xy) (xy != 1)`

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:

y – cos y = x :  (y sin y + cos y + x) y′ = y

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:

x + y = tan^–1y   :   y² y′ + y² + 1 = 0

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:

`y = sqrt(a^2 - x^2 )  x in (-a,a) : x + y  dy/dx = 0(y != 0)`

The number of arbitrary constants in the general solution of a differential equation of fourth order are ______.

The number of arbitrary constants in the particular solution of a differential equation of third order are ______.

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

`x/a + y/b = 1`

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

y² = a (b² – x²)

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

y = a e^3x + b e^{– 2x}

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

y = e^2x (a + bx)

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

y = e^x (a cos x + b sin x)

Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.

Form the differential equation of the family of hyperbolas having foci on x-axis and centre at origin.

Which of the following differential equations has y = c₁ e^x + c₂ e^–x as the general solution?

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

(B) `(d^2y)/(dx^2) - y = 0`

(C) `(d^2y)/(dx^2) + 1 = 0`

(D) `(d^2y)/(dx^2)  - 1 = 0`

Which of the following differential equation has y = x as one of its particular solution?

A. `(d^2y)/(dx^2) - x^2 (dy)/(dx) + xy = x`

B. `(d^2y)/(dx^2) + x dy/dx + xy = x`

C. `(d^2y)/(dx^2) - x^2 dy/dx + xy = 0`

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

For the differential equation, find the general solution:

`dy/dx = (1 - cos x)/(1+cos x)`

For the differential equation, find the general solution:

`dy/dx = sqrt(4-y^2)      (-2 < y < 2)`

For the differential equation, find the general solution:

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

For the differential equation, find the general solution:

sec² x tan y dx + sec² y tan x dy = 0

For the differential equation, find the general solution:

(e^x + e^–x) dy – (e^x – e–x) dx = 0

For the differential equation, find the general solution:

`dy/dx = (1+x^2)(1+y^2)`

For the differential equation, find the general solution:

y log y dx - x dy = 0

For the differential equation, find the general solution:

`x^5  dy/dx = - y^5`

For the differential equation, find the general solution:

`dy/dx = sin^(-1) x`

For the differential equation, find the general solution:

e^x tan y dx + (1 – e^x) sec² y dy = 0

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

`(x^3 + x^2 + x + 1) dy/dx = 2x^2 + x; y = 1` When x = 0

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

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

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

`cos (dx/dy) = a(a in R); y = 1` when x = 0

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

`dy/dx` = y tan x; y = 1 when x = 0

Find the equation of a curve passing through the point (0, 0) and whose differential equation is y′ = e ^x sin x.

For the differential equation `xy(dy)/(dx) = (x + 2)(y + 2)`  find the solution curve passing through the point (1, –1).

Find the equation of a curve passing through the point (0, -2) given that at any point (x, y) on the curve, the product of the slope of its tangent and y-coordinate of the point is equal to the x-coordinate of the point.

At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (- 4, -3). Find the equation of the curve given that it passes through (-2, 1).

The volume of 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 balloon after t seconds.

In a bank, principal increases continuously at the rate of r% per year. Find the value of r if Rs 100 doubles itself in 10 years (log­_e 2 = 0.6931).

In a bank, principal increases continuously at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years (e^0.5 = 1.648).

In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?

The general solution of the differential equation `dy/dx = e^(x+y)` is ______.

Show that the given differential equation is homogeneous and solve them.

(x² + xy) dy = (x² + y²) dx

Show that the given differential equation is homogeneous and solve them.

`y' = (x + y)/x`

Show that the given differential equation is homogeneous and solve them.

(x – y) dy – (x + y) dx = 0

Show that the given differential equation is homogeneous and solve them.

(x² – y²) dx + 2xy dy = 0

Show that the given differential equation is homogeneous and solve them.

`x^2 dy/dx = x^2 - 2y^2 + xy`

Show that the given differential equation is homogeneous and solve them.

`x  dy - y  dx =  sqrt(x^2 + y^2)   dx`

Show that the given differential equation is homogeneous and solve them.

`{xcos(y/x) + ysin(y/x)}ydx = {ysin (y/x) -  xcos(y/x)}xdy`

Show that the given differential equation is homogeneous and solve them.

`x dy/dx - y +  x sin (y/x) = 0`

Show that the given differential equation is homogeneous and solve them.

`y  dx + x log(y/x)dy - 2x  dy = 0`

Show that the given differential equation is homogeneous and solve them.

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

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

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

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

x² dy + (xy + y²) dx = 0; y = 1 when x = 1

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

`[xsin^2(y/x - y)] dx + x  dy = 0; y = pi/4 "when"  x = 1`

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

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

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

`2xy + y^2 - 2x^2  dy/dx = 0; y = 2`   when x  = 1

A homogeneous differential equation of the from `dx/dy = h (x/y)` can be solved by making the substitution.

Which of the following is a homogeneous differential equation?

For the differential equation, find the general solution:

`dy/dx  + 2y = sin x`

For the differential equation, find the general solution:

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

For the differential equation, find the general solution:

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

For the differential equation, find the general solution:

`dy/dx + (sec x) y = tan x (0 <= x < pi/2)`

For the differential equation, find the general solution:

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

For the differential equation, find the general solution:

`x dy/dx +  2y= x^2 log x`

For the differential equation, find the general solution:

`x log x dy/dx + y=    2/x log x`

For the differential equation, find the general solution:

(1 + x²) dy + 2xy dx = cot x dx (x ≠ 0)

For the differential equation, find the general solution:

`x dy/dx + y - x + xy cot x = 0(x != 0)`

For the differential equation, find the general solution:

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

For the differential equation, find the general solution:

y dx + (x – y²) dy = 0

For the differential equation, find the general solution:

`(x + 3y^2) dy/dx = y(y > 0)`

For the differential equation given, find a particular solution satisfying the given condition:

`dy/dx + 2y tan x = sin x; y = 0 " when x " = pi/3`

For the differential equation given, find a particular solution satisfying the given condition:

`(1 + x^2)dy/dx + 2xy = 1/(1 + x^2); y = 0`  when x = 1

For the differential equation given, find a particular solution satisfying the given condition:

`dy/dx - 3ycotx = sin 2x; y = 2`  when `x = pi/2`

Find the equation of the curve passing through the origin given that the slope of the tangent to the curve at any point (x, y) is equal to the sum of the coordinates of the point.

Find the equation of a curve passing through the point (0, 2) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.

The Integrating Factor of the differential equation `dy/dx - y = 2x^2` is ______.

The integrating factor of the differential equation.

`(1 - y^2) dx/dy + yx = ay(-1 < y < 1)` is ______.

For the differential equation given below, indicate its order and degree (if defined).

`(d^2y)/dx^2 + 5x(dy/dx)^2 - 6y = log x`

For the differential equation given below, indicate its order and degree (if defined).

`((dy)/(dx))^3 -4(dy/dx)^2 + 7y = sin x`

For the differential equation given below, indicate its order and degree (if defined).

`(d^4y)/dx^4 - sin ((d^3y)/(dx^3)) = 0`

For the given below, verify that the given function (implicit or explicit) is a solution to the corresponding differential equation.

xy = a e^x + b e^-x + x² : `x (d^2y)/(dx^2) + 2 dy/dx - xy + x^2 - 2 = 0`

For the given below, verify that the given function (implicit or explicit) is a solution to the corresponding differential equation.

`y = e^x (acos x + b sin x)  :  (d^2y)/(dx^2) - 2 dy/dx + 2y = 0`

For the given below, verify that the given function (implicit or explicit) is a solution to the corresponding differential equation.

`y = xsin 3x   :   (d^2y)/(dx^2) + 9y - 6 cos 3x = 0`

For the given below, verify that the given function (implicit or explicit) is a solution to the corresponding differential equation.

`x^2 = 2y^2 log y : (x^2  + y^2) dy/dx - xy = 0`

Form the differential equation representing the family of curves given by (x – a)² + 2y² = a², where a is an arbitrary constant.

Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.

Prove that x² – y² = c (x² + y²)² is the general solution of differential equation  (x³ – 3x y²) dx = (y³ – 3x²y) dy, where c is a parameter.

Find the general solution of the differential equation `dy/dx + sqrt((1-y^2)/(1-x^2)) = 0.`

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

Find the equation of the curve passing through the point `(0,pi/4)`, whose differential equation is sin x cos y dx + cos x sin y dy = 0.

Find the particular solution of the differential equation (1 + e^2x) dy + (1 + y²) e^x dx = 0, given that y = 1 when x = 0.

Solve the differential equation  `ye^(x/y) dx = (xe^(x/y) + y^2)dy, (y != 0)`

Find a particular solution of the differential equation (x - y) (dx + dy) = dx - dy, given that y = -1, when x = 0. (Hint: put x - y = t)

Solve the differential equation `[e^(-2sqrtx)/sqrtx - y/sqrtx] dx/dy = 1 (x != 0).`

Find a particular solution of the differential equation `dy/dx + y cot x = 4xcosec x(x != 0)`, given that y = 0 when `x = pi/2.`

The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?

The general solution of the differential equation `(y dx - x dy)/y = 0` is ______.

The general solution of a differential equation of the type  `dx/dy + P_1 x = Q_1` is ______.

The general solution of the differential equation e^x dy + (y e^x + 2x) dx = 0 is ______.