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

If the straight lines  \[\frac{x - 1}{2} = \frac{y + 1}{k} = \frac{z}{2}\] and \[\frac{x + 1}{2} = \frac{y + 1}{2} = \frac{z}{k}\] are coplanar, find the equations of the planes containing them.

Verify that xy = a e^x + b e^−x + x² is a solution of the differential equation \[x\frac{d^2 y}{d x^2} + 2\frac{dy}{dx} - xy + x^2 - 2 = 0.\]

Show that y = C x + 2C² is a solution of the differential equation \[2 \left( \frac{dy}{dx} \right)^2 + x\frac{dy}{dx} - y = 0.\]

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