Science (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

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 ______.

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`

Show that the function given by f(x) = sin x is

1. strictly increasing in `(0, pi/2)`
2. strictly decreasing in `(pi/2, pi)`
3. neither increasing nor decreasing in (0, π)

Find the intervals in which the function f given by f(x) = 2x² − 3x is

1. strictly increasing
2. strictly decreasing

Find the intervals in which the function f given by f(x) = 2x³ − 3x² − 36x + 7 is

1. Strictly increasing
2. Strictly decreasing

Find the intervals in which the following functions are strictly increasing or decreasing:

x² + 2x − 5

Find the intervals in which the following functions are strictly increasing or decreasing:

10 − 6x − 2x²