Let F denotes the family of ellipses whose centre is at the origin and major axis is the y-axis. Then, equation of the family F is :

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \frac{\mathrm{dy}}{\mathrm{dx}}\left(\mathrm{x}\frac{{\displaystyle \mathrm{dy}}}{{\displaystyle \mathrm{dx}}} - \mathrm{y}\right) = 0$

• $\mathrm{xy}\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \frac{\mathrm{dy}}{\mathrm{dx}}\left(\mathrm{x}\frac{{\displaystyle \mathrm{dy}}}{{\displaystyle \mathrm{dx}}} - \mathrm{y}\right) = 0$

• $\mathrm{xy}\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \frac{\mathrm{dy}}{\mathrm{dx}}\left(\mathrm{x}\frac{{\displaystyle \mathrm{dy}}}{{\displaystyle \mathrm{dx}}} - \mathrm{y}\right) = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - \frac{\mathrm{dy}}{\mathrm{dx}}\left(\mathrm{x}\frac{{\displaystyle \mathrm{dy}}}{{\displaystyle \mathrm{dx}}} - \mathrm{y}\right) = 0$

Solution of the equation $\mathrm{x}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 + 2\sqrt{\mathrm{xy}}\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{y} = 0$ is :

• x + y = a

• $\sqrt{\mathrm{x}} - \sqrt{\mathrm{y}} = \sqrt{\mathrm{a}}$

• x² + y² = a²

• $\sqrt{\mathrm{x}} + \sqrt{\mathrm{y}} = \sqrt{\mathrm{a}}$

The solution of differential equation (x + y )(dx - dy) = dx + dy is :

• x - y = ke^{x - y}

• x + y = ke^{x + y}

• x + y = k(x - y)

• x + y = ke^{x - y}

The solution of $\frac{\mathrm{dy}}{\mathrm{dx}} + 1 = \csc \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right)$ is  :

• $\cos \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right) +\hspace{0.17em}\mathrm{x} = \mathrm{c}$

• $\cos \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right) = \mathrm{c}$

• $\sin \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right) + \mathrm{x} = \mathrm{c}$

• $\sin \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right) + \sin \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right) = \mathrm{c}$

The order of the differential equation

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

• 2

• 3

• $\frac{1}{2}$

• 4

496.

The integrating factor of the differential equation $\cos \left(\mathrm{x}\right)\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) + \mathrm{ysin}\left(\mathrm{x}\right)$ = 1 is :

• $\cos \left(\mathrm{x}\right)$

• $\tan \left(\mathrm{x}\right)$

• $\sin \left(\mathrm{x}\right)$

• $\sec \left(\mathrm{x}\right)$

Solution of the differential equation tan(y) . sec²(x)dx + tan(x) · sec²(y)dy = 0 is

• tan(x) + tan(y) = k

• tan(x) - tan(y) = k

• $\left(\raisebox{1ex}{$\tan \left(\mathrm{x}\right)$}\!\left/ \!\raisebox{-1ex}{$\tan \left(\mathrm{y}\right)$}\right.\right) = \mathrm{k}$

• $\tan \left(\mathrm{x}\right) . \tan \left(\mathrm{y}\right) = \mathrm{k}$

The differential equation of all non-horizontal lines in a plane is :

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

• $\frac{\mathrm{dx}}{\mathrm{dy}}$ = 0

• $\frac{\mathrm{dy}}{\mathrm{dx}}$ = 0

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

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

• 2, 2

• 2, 1

• 1, 2

• 2, 3

The solution of 2(y + 3) - xy $\frac{\mathrm{dy}}{\mathrm{dx}}$ = 0 with y = - 2,when x = 1 is

• (y + 3) = x²

• x²(y + 3) = 1

• x⁴(y + 3) = 1

• x²(y + 3)³ = e^{y + 2}