The number of real solutions of ${\tan }^{-1}\left(\sqrt{\mathrm{x}\left(\mathrm{x} + 1\right)}\right) + {\sin }^{-1}\left(\sqrt{{\mathrm{x}}^2 + \mathrm{x} + 1}\right) = \frac{\mathrm{\pi }}{2}$ is

• zero

• one

• two

• infinite

The differential equation of all straight lines passing through the point (1, - 1), is

• $\mathrm{y} = \left(\mathrm{x} + 1\right)\frac{\mathrm{dy}}{\mathrm{dx}} + 1$

• $\mathrm{y} = \left(\mathrm{x} + 1\right)\frac{\mathrm{dy}}{\mathrm{dx}} - 1$

• $\mathrm{y} = \left(\mathrm{x} - 1\right)\frac{\mathrm{dy}}{\mathrm{dx}} + 1$

• $\mathrm{y} = \left(\mathrm{x} - 1\right)\frac{\mathrm{dy}}{\mathrm{dx}} - 1$

Integrating factor of differential equation $\cos \left(\mathrm{x}\right)\frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{ysin}\left(\mathrm{x}\right) = 1$ is

• $\cos \left(\mathrm{x}\right)$

• $\tan \left(\mathrm{x}\right)$

• $\sec \left(\mathrm{x}\right)$

• $\sin \left(\mathrm{x}\right)$

The order and degree of the differential equation $\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right) + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} + {\mathrm{x}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.} = 0$ are respectively

• 2, 3

• 3, 3

• 2, 6

• 2, 4

635.

The general solution of the differential equation ydx + (1 + x²) tan^{- 1}(x)dy = 0, is

• ${\mathrm{ytan}}^{-1}\left(\mathrm{x}\right) = \mathrm{c}$

• ${\mathrm{xtan}}^{-1}\left(\mathrm{y}\right) = \mathrm{c}$

• $\mathrm{y} + {\tan }^{-1}\left(\mathrm{x}\right) = \mathrm{c}$

• $\mathrm{x} + {\tan }^{-1}\left(\mathrm{y}\right) = \mathrm{c}$

Solution of differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = 2\mathrm{xy}$ is

• y = ${\mathrm{ce}}^{{\mathrm{x}}^2}$

• y² = 2x² + c

• y = ${\mathrm{ce}}^{- {\mathrm{x}}^2}$

• y = x² + c

The second order differential equation is

• y' + x = y²

• y'y'' + y = sin(x)

• y''' + y'' + y = 0

• y' = y

The solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} - \frac{\mathrm{y}}{\mathrm{x}} = 1$ is

• x²log_e(x) + y = c

• xlog_e(x) + cx = y

• x²log_e(x) - y = c

• xlog_e(x) + y = cx

An integrating factor of the differential equation $\left(1 - {\mathrm{x}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} - \mathrm{xy} = 1$, is

• - x

• $\frac{\mathrm{x}}{\left(1 - {\mathrm{x}}^2\right)}$

• $\sqrt{\left(1 - {\mathrm{x}}^2\right)}$

• $\frac{1}{2}\log \left(1 - {\mathrm{x}}^2\right)$

Solve $\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)\tan \left(\mathrm{y}\right) = \sin \left(\mathrm{x} + \mathrm{y}\right) + \sin \left(\mathrm{x} - \mathrm{y}\right)$

• $\sec \left(\mathrm{x}\right) - \frac{1}{2}\tan \left(\mathrm{y}\right) = \mathrm{c}$

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

• $\sec \left(\mathrm{x}\right) + \tan \left(\mathrm{y}\right) = \mathrm{c}$

• $\sec \left(\mathrm{y}\right) + 2\cos \left(\mathrm{x}\right) = \mathrm{c}$