51.

Solution of the differential equation xdy - ydx = 0 represents a

• parabola

• circle

• hyperbola

• straight line

The general solution of the differential equation

$\frac{d\mathrm{y}}{d\mathrm{x}} = {\mathrm{e}}^{\mathrm{y} + \mathrm{x}} + {\mathrm{e}}^{\mathrm{y} - \mathrm{x}}$ is

• e^{- y} = e^x - e^{- x} + c

• e^{- y} = e^{- x} - e^x + c

• e^{- y} = e^x + e^- x + c

• e^y = e^x + e^{- x} + c

The order of the differential equation

$\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} = \sqrt{1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2}$ is

• 3

• 2

• 1

• 4

The degree of the differential equation

${\left[1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right]}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} = \frac{d^2\mathrm{y}}{d{\mathrm{x}}^2}$ is

• 1

• 5

• $\frac{10}{3}$

• 3

The differential equation of all parabolas whose axes are parallel to y-axis, is

• $\frac{d^3\mathrm{y}}{d{\mathrm{x}}^3} = 0$

• $\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} = 0$

• $\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} + \frac{d\mathrm{y}}{d\mathrm{x}} = 0$

• $\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} + \frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{y} = 0$

The solution of the differential equation $\frac{d\mathrm{y}}{d\mathrm{x}} = {\mathrm{e}}^{\mathrm{y} + \mathrm{x}} + {\mathrm{e}}^{\mathrm{y} - \mathrm{x}}$ is

• e^{- y} = e^x - e^{- x} + c, c integrating constant

• e^{- y} = e^- x - e^x + c, c integrating constant

• e^{- y} = e^x + e^{- x} + c, c integrating constant

• e^{- y} + e^x - e^{- x} = c, c integrating constant

If $\mathrm{x} = {\mathrm{e}}^{\mathrm{t}}\sin \left(\mathrm{t}\right), \mathrm{y} = {\mathrm{e}}^{\mathrm{t}}\cos \left(\mathrm{t}\right)$, then $\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2}$ at $\mathrm{x} = \mathrm{\pi }$ is

• $2{\mathrm{e}}^{\mathrm{\pi }}$

• $\frac{1}{2}{\mathrm{e}}^{\mathrm{\pi }}$

• $\frac{1}{2{\mathrm{e}}^{\mathrm{\pi }}}$

• $\frac{2}{{\mathrm{e}}^{\mathrm{\pi }}}$

The value of $\frac{d\mathrm{y}}{d\mathrm{x}} \mathrm{at} \mathrm{x} = \frac{\mathrm{\pi }}{2}$, where y is given by $\mathrm{y} = {\mathrm{x}}^{\sin \left(\mathrm{x}\right)} + \sqrt{\mathrm{x}}$, is

• $1 + \frac{1}{\sqrt{2\mathrm{\pi }}}$

• 1

• $\frac{1}{\sqrt{2\mathrm{\pi }}}$

• $1 - \frac{1}{\sqrt{2\mathrm{\pi }}}$

The order and degree of the following differential equation ${\left[1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right]}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} = \frac{d^3\mathrm{y}}{d{\mathrm{x}}^3}$ are respectively

• 3, 2

• 3, 10

• 2, 3

• 3, 5

The differential equation of the family of circles passing through the fixed points (a, 0) and (- a, 0) is

• y₁(y² - x²) + 2xy + a² = 0

• y₁y² + xy + a²x² = 0

• y₁(y² - x² + a²) + 2xy = 0

• y₁(y² + x²) - 2xy + a² =  0