NCERT Exemplar solutions for माठेमटिक्स एक्सेम्पलार [इंग्रजी] इयत्ता १२ chapter 9 - Differential Equations [Latest edition]

Find the differential equation of the family of curves y = Ae^2x + B.e^–2x.

Find the general solution of the differential equation `"dy"/"dx" = y/x`.

Given that `"dy"/"dx"` = ye^x and x = 0, y = e. Find the value of y when x = 1.

Solve the differential equation `"dy"/"dx" + y/x` = x².

Find the differential equation of the family of lines through the origin.

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

Find the equation of a curve whose tangent at any point on it, different from origin, has slope `y + y/x`.

Find the equation of a curve passing through the point (1, 1) if the perpendicular distance of the origin from the normal at any point P(x, y) of the curve is equal to the distance of P from the x-axis.

Find the equation of a curve passing through `(1, pi/4)` if the slope of the tangent to the curve at any point P(x, y) is `y/x - cos^2  y/x`.

Solve `x^2 "dy"/"dx" - xy = 1 + cos(y/x)`, x ≠ 0 and x = 1, y = `pi/2`

State the type of the differential equation for the equation. xdy – ydx = `sqrt(x^2 + y^2)  "d"x` and solve it

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

The degree of the differential equation `("d"^2y)/("d"x^2) + 3("dy"/"dx")^2 = x^2 log(("d"^2y)/("d"x^2))` is ______.

The order and degree of the differential equation `[1 + ("dy"/"dx")^2]^2 = ("d"^2y)/("d"x^2)` respectively, are ______.

The order of the differential equation of all circles of given radius a is ______.

The solution of the differential equation `2x * "dy"/"dx" y` = 3 represents a family of ______.

The integrating factor of the differential equation `"dy"/"dx" (x log x) + y` = 2logx is ______.

A solution of the differential equation `("dy"/"dx")^2 - x "dy"/"dx" + y` = 0 is ______.

Which of the following is not a homogeneous function of x and y.

Solution of the differential equation `"dx"/x + "dy"/y` = 0 is ______.

The solution of the differential equation `x "dt"/"dx" + 2y` = x² is ______.

Order of the differential equation representing the family of parabolas y² = 4ax is ______.

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

The number of arbitrary constants in a particular solution of the differential equation tan x dx + tan y dy = 0 is ______.

F(x, y) = `(sqrt(x^2 + y^2) + y)/x` is a homogeneous function of degree ______.

An appropriate substitution to solve the differential equation `"dx"/"dy" = (x^2 log(x/y) - x^2)/(xy log(x/y))` is ______.

Integrating factor of the differential equation `x "dy"/"dx" - y` = sinx is ______.

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

The general solution of the differential equation `"dy"/"dx" + y/x` = 1 is ______.

The differential equation representing the family of curves y = A sinx + B cosx is ______.

`("e"^(-2sqrt(x))/sqrt(x) - y/sqrt(x))("d"x)/("d"y) = 1(x ≠ 0)` when written in the form `"dy"/"dx" + "P"y` = Q, then P = ______.

Order of the differential equation representing the family of ellipses having centre at origin and foci on x-axis is two.

Degree of the differential equation `sqrt(1 + ("d"^2y)/("d"x^2)) = x + "dy"/"dx"` is not defined.

`"dy"/"dx" + y` = 5 is a differential equation of the type `"dy"/"dx" + "P"y` = Q but it can be solved using variable separable method also.

F(x, y) = `(ycos(y/x) + x)/(xcos(y/x))` is not a homogeneous function.

F(x, y) = `(x^2 + y^2)/(x - y)` is a homogeneous function of degree 1.

Integrating factor of the differential equation `"dy"/"dx" - y` = cos x is e^x.

The general solution of the differential equation x(1 + y²)dx + y(1 + x²)dy = 0 is (1 + x²)(1 + y²) = k.

The general solution of the differential equation `"dy"/"dx" + y sec x` = tan x is y(secx – tanx) = secx – tanx + x + k.

x + y = tan^–1y is a solution of the differential equation `y^2 "dy"/"dx" + y^2 + 1` = 0.

y = x is a particular solution of the differential equation `("d"^2y)/("d"x^2) - x^2 "dy"/"dx" + xy` = x.

Find the solution of `"dy"/"dx"` = 2^y–x.

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

Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.

Solve the differential equation `(x^2 - 1) "dy"/"dx" + 2xy = 1/(x^2 - 1)`.

Solve the differential equation `"dy"/"dx" + 2xy` = y

Find the general solution of `"dy"/"dx" + "a"y` = e^mx 

Solve the differential equation `"dy"/"dx" + 1` = e^{x + y}.

Solve: ydx – xdy = x²ydx.

Solve the differential equation `"dy"/"dx"` = 1 + x + y² + xy², when y = 0, x = 0.

Find the general solution of `(x + 2y^3)  "dy"/"dx"` = y

If y(x) is a solution of `((2 + sinx)/(1 + y))"dy"/"dx"` = – cosx and y (0) = 1, then find the value of `y(pi/2)`.

If y(t) is a solution of `(1 + "t")"dy"/"dt" - "t"y` = 1 and y(0) = – 1, then show that y(1) = `-1/2`.

Form the differential equation having y = (sin^–1x)² + Acos^–1x + B, where A and B are arbitrary constants, as its general solution.

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

Find the equation of a curve passing through origin and satisfying the differential equation `(1 + x^2) "dy"/"dx" + 2xy` = 4x² 

Solve : `x^2 "dy"/"dx"` = x² + xy + y².

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

Find the general solution of y²dx + (x² – xy + y²) dy = 0.

Solve: (x + y)(dx – dy) = dx + dy. [Hint: Substitute x + y = z after seperating dx and dy]

Solve:

`2(y + 3) - xy  (dy)/(dx)` = 0, given that y(1) = – 2.

Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`

Form the differential equation by eliminating A and B in Ax² + By² = 1

Solve the differential equation (1 + y²) tan^–1xdx + 2y(1 + x²)dy = 0.

Find the differential equation of system of concentric circles with centre (1, 2).

Solve: `y + "d"/("d"x) (xy) = x(sinx + logx)`

Find the general solution of (1 + tany)(dx – dy) + 2xdy = 0.

Solve: `("d"y)/("d"x) = cos(x + y) + sin(x + y)`. [Hint: Substitute x + y = z]

Find the general solution of `("d"y)/("d"x) -3y = sin2x`

Find the equation of a curve passing through (2, 1) if the slope of the tangent to the curve at any point (x, y) is `(x^2 + y^2)/(2xy)`.

Find the equation of the curve through the point (1, 0) if the slope of the tangent to the curve at any point (x, y) is `(y - 1)/(x^2 + x)`

Find the equation of a curve passing through origin if the slope of the tangent to the curve at any point (x, y) is equal to the square of the difference of the abcissa and ordinate of the point.

Find the equation of a curve passing through the point (1, 1). If the tangent drawn at any point P(x, y) on the curve meets the co-ordinate axes at A and B such that P is the mid-point of AB.

Solcve: `x ("d"y)/("d"x) = y(log y – log x + 1)`

The degree of the differential equation `(("d"^2y)/("d"x^2))^2 + (("d"y)/("d"x))^2 = xsin(("d"y)/("d"x))` is ______.

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

The order and degree of the differential equation `("d"^2y)/("d"x^2) + (("d"y)/("d"x))^(1/4) + x^(1/5)` = 0, respectively, are ______.

If y = e^–x (Acosx + Bsinx), then y is a solution of ______.

The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is ______.

Solution of differential equation xdy – ydx = 0 represents : ______.

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

Solution of the differential equation tany sec²xdx + tanx sec²ydy = 0 is ______.

Family y = Ax + A³ of curves is represented by the differential equation of degree ______.

Integrating factor of `(x"d"y)/("d"x) - y = x^4 - 3x` is ______.

Solution of `("d"y)/("d"x) - y` = 1, y(0) = 1 is given by ______.

The number of solutions of `("d"y)/("d"x) = (y + 1)/(x - 1)` when y (1) = 2 is ______. 

Which of the following is a second order differential equation?

Integrating factor of the differential equation `(1 - x^2) ("d"y)/("d"x) - xy` = 1 is ______.

tan^–1x + tan^–1y = c is the general solution of the differential equation ______.

The differential equation `y ("d"y)/("d"x) + "c"` represents: ______.

The general solution of e^x cosy dx – e^x siny dy = 0 is ______.

The degree of the differential equation `("d"^2y)/("d"x^2) + (("d"y)/("d"x))^3 + 6y^5` = 0 is ______.

The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.

Integrating factor of the differential equation `("d"y)/("d"x) + y tanx - secx` = 0 is ______.

The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is ______.

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

y = ae^mx+ be^–mx satisfies which of the following differential equation?

The solution of the differential equation cosx siny dx + sinx cosy dy = 0 is ______.

The solution of `x ("d"y)/("d"x) + y` = e^x is ______.

The differential equation of the family of curves x² + y² – 2ay = 0, where a is arbitrary constant, is ______.

Family y = Ax + A³ of curves will correspond to a differential equation of order ______.

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

The curve for which the slope of the tangent at any point is equal to the ratio of the abcissa to the ordinate of the point is ______.

The general solution of the differential equation `("d"y)/("d"x) = "e"^(x^2/2) + xy` is ______.

The solution of the equation (2y – 1)dx – (2x + 3)dy = 0 is ______.

The differential equation for which y = acosx + bsinx is a solution, is ______.

The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.

The order and degree of the differential equation `(("d"^3y)/("d"x^3))^2 - 3 ("d"^2y)/("d"x^2) + 2(("d"y)/("d"x))^4` = y⁴ are ______.

The order and degree of the differential equation `[1 + ((dy)/(dx))^2] = (d^2y)/(dx^2)` are ______.

The differential equation of the family of curves y² = 4a(x + a) is ______.

Which of the following is the general solution of `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + y` = 0?

General solution of `("d"y)/("d"x) + ytanx = secx` is ______.

Solution of the differential equation `("d"y)/("d"x) + y/x` = sec x is ______.

The general solution of the differential equation (e^x + 1) ydy = (y + 1) e^xdx is ______.

The solution of the differential equation `("d"y)/("d"x) = "e"^(x - y) + x^2 "e"^-y` is ______.

The solution of the differential equation `("d"y)/("d"x) + (2xy)/(1 + x^2) = 1/(1 + x^2)^2` is ______.

The degree of the differential equation `("d"^2y)/("d"x^2) + "e"^((dy)/(dx))` = 0 is ______.

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

The number of arbitrary constants in the general solution of a differential equation of order three is ______.

`("d"y)/("d"x) + y/(xlogx) = 1/x` is an equation of the type ______.

General solution of the differential equation of the type `("d"x)/("d"x) + "P"_1x = "Q"_1` is given by ______.

The solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is ______.

The solution of `(1 + x^2) ("d"y)/("d"x) + 2xy - 4x^2` = 0 is ______.

The solution of the differential equation ydx + (x + xy)dy = 0 is ______.

General solution of `("d"y)/("d"x) + y` = sinx is ______.

The solution of differential equation coty dx = xdy is ______.

The integrating factor of `("d"y)/("d"x) + y = (1 + y)/x` is ______.

Integrating factor of the differential equation of the form `("d"x)/("d"y) + "P"_1x = "Q"_1` is given by `"e"^(int P_1dy)`.

Solution of the differential equation of the type `("d"x)/("d"y) + "p"_1x = "Q"_1` is given by x.I.F. = `("I"."F") xx "Q"_1"d"y`.

Correct substitution for the solution of the differential equation of the type `("d"y)/("d"x) = "f"(x, y)`, where f(x, y) is a homogeneous function of zero degree is y = vx.

Correct substitution for the solution of the differential equation of the type `("d"x)/("d"y) = "g"(x, y)` where g(x, y) is a homogeneous function of the degree zero is x = vy.

Number of arbitrary constants in the particular solution of a differential equation of order two is two.

The differential equation representing the family of circles x² + (y – a)² = a² will be of order two.

The solution of `("d"y)/("d"x) = (y/x)^(1/3)` is `y^(2/3) - x^(2/3)` = c.

Differential equation representing the family of curves y = e^x (Acosx + Bsinx) is `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + 2y` = 0

The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx².

Solution of `x("d"y)/("d"x) = y + x tan  y/x` is `sin(y/x)` = cx

The differential equation of all non horizontal lines in a plane is `("d"^2x)/("d"y^2)` = 0