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

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
x² + y² = a²

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
x² − y² = a²

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y² = 4ax

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
x² + (y − b)² = 1

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
(x − a)² − y² = 1

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y² = 4a (x − b)

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y = ax³

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
x² + y² = ax³

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y = e^ax

Show that y = be^x + ce^2x is a solution of the differential equation, \[\frac{d^2 y}{d x^2} - 3\frac{dy}{dx} + 2y = 0\]

Find the equation of a curve passing through the point (0, 0) and whose differential equation is \[\frac{dy}{dx} = e^x \sin x\]

For the differential equation xy \[\frac{dy}{dx}\] = (x + 2) (y + 2). Find the solution curve passing through the point (1, −1).

Find one-parameter families of solution curves of the following differential equation:-

\[\frac{dy}{dx} + 3y = e^{mx}\], m is a given real number.

Find one-parameter families of solution curves of the following differential equation:-

\[\frac{dy}{dx} - y = \cos 2x\]

Find one-parameter families of solution curves of the following differential equation:-

\[x\frac{dy}{dx} - y = \left( x + 1 \right) e^{- x}\]

Find one-parameter families of solution curves of the following differential equation:-

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

Find one-parameter families of solution curves of the following differential equation:-

\[\left( x \log x \right)\frac{dy}{dx} + y = \log x\]

Find one-parameter families of solution curves of the following differential equation:-

\[\frac{dy}{dx} - \frac{2xy}{1 + x^2} = x^2 + 2\]