Science (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

Solve the following initial value problem:
(y⁴ − 2x³ y) dx + (x⁴ − 2xy³) dy = 0, y (1) = 1

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?

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"dx"/sqrt(5-4"x" - 2"x"^2)`

Integrate the function `cos("x + a")/sin("x + b")` w.r.t. x.

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

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