PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

Show that y² − x² − xy = a is a solution of the differential equation \[\left( x - 2y \right)\frac{dy}{dx} + 2x + y = 0.\]

Verify that y = A cos x + sin x satisfies the differential equation \[\cos x\frac{dy}{dx} + \left( \sin x \right)y=1.\]

Find the differential equation corresponding to y = ae^2x + be^−3x + ce^x where a, b, c are arbitrary constants.

Show that the differential equation of all parabolas which have their axes parallel to y-axis is \[\frac{d^3 y}{d x^3} = 0.\]

From x² + y² + 2ax + 2by + c = 0, derive a differential equation not containing a, b and c.

\[\frac{dy}{dx} = \sin^3 x \cos^4 x + x\sqrt{x + 1}\]

\[\frac{dy}{dx} = \frac{1}{x^2 + 4x + 5}\]

\[\frac{dy}{dx} = y^2 + 2y + 2\]

\[\frac{dy}{dx} + 4x = e^x\]

\[\frac{dy}{dx} = x^2 e^x\]

\[\frac{dy}{dx} - x \sin^2 x = \frac{1}{x \log x}\]

\[(\tan^2 x + 2\tan x + 5)\frac{dy}{dx} = 2(1+\tan x)\sec^2x\]

\[\frac{dy}{dx} = \sin^3 x \cos^2 x + x e^x\]

tan y dx + tan x dy = 0

(1 + x) y dx + (1 + y) x dy = 0

x cos² y dx = y cos² x dy

cos y log (sec x + tan x) dx = cos x log (sec y + tan y) dy

cosec x (log y) dy + x²y dx = 0

(1 − x²) dy + xy dx = xy² dx

Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx