Commerce (English Medium) इयत्ता १२ - CBSE Question Bank Solutions

Solve the following initial value problem:
x (x² + 3y²) dx + y (y² + 3x²) dy = 0, y (1) = 1

Solve the following initial value problem:
\[\left\{ x \sin^2 \left( \frac{y}{x} \right) - y \right\}dx + x dy = 0, y\left( 1 \right) = \frac{\pi}{4}\]

Solve the following initial value problem:
\[x\frac{dy}{dx} - y + x \sin\left( \frac{y}{x} \right) = 0, y\left( 2 \right) = x\]

Find the particular solution of the differential equation x cos\[\left( \frac{y}{x} \right)\frac{dy}{dx} = y \cos\left( \frac{y}{x} \right) + x\], given that when x = 1, \[y = \frac{\pi}{4}\]

Find the particular solution of the differential equation \[\left( x - y \right)\frac{dy}{dx} = x + 2y\], given that when x = 1, y = 0.

Show that the family of curves for which \[\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}\], is given by \[x^2 - y^2 = Cx\]

A homogeneous differential equation of the form \[\frac{dx}{dy} = h\left( \frac{x}{y} \right)\] can be solved by making the substitution

Which of the following is a homogeneous differential equation?

Differentiate : \[\tan^{- 1} \left( \frac{1 + \cos x}{\sin x} \right)\] with respect to x .

Solve the following differential equation : \[\left[ y - x  \cos\left( \frac{y}{x} \right) \right]dy + \left[ y  \cos\left( \frac{y}{x} \right) - 2x  \sin\left( \frac{y}{x} \right) \right]dx = 0\] .

Evaluate : \[\int\limits_0^\pi \frac{x \tan x}{\sec x \cdot cosec x}dx\] .

Solve the differential equation:  ` (dy)/(dx) = (x + y )/ (x - y )`

Find `int_  (sin "x" - cos "x" )/sqrt(1 + sin 2"x") d"x", 0 < "x" < π / 2 `

Find `int_  sin ("x" - a)/(sin ("x" + a )) d"x"`

Find `int_  (log "x")^2 d"x"`

Find `int_  (sin2"x")/((sin^2 "x"+1)(sin^2"x"+3))d"x"`

Find the area of the triangle whose vertices are (-1, 1), (0, 5) and (3, 2), using integration. 

Find: `int_  (cos"x")/((1 + sin "x") (2+ sin"x")) "dx"`

Find: `int sec^2 x /sqrt(tan^2 x+4) dx.`

Find: `intsqrt(1 - sin 2x) dx, pi/4 < x < pi/2`