The solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = 2{\mathrm{e}}^{\mathrm{x} - \mathrm{y}} + {\mathrm{x}}^2{\mathrm{e}}^{- \mathrm{y}}$ is

• ${\mathrm{e}}^{\mathrm{y}} = 2{\mathrm{e}}^{\mathrm{x}} + \frac{{\mathrm{x}}^3}{3} + \mathrm{C}$

• ${\mathrm{e}}^{- \mathrm{y}} = 2{\mathrm{e}}^{\mathrm{x}} + \frac{{\mathrm{x}}^{- 3}}{3} + \mathrm{C}$

• ${\mathrm{e}}^{- \mathrm{y}} = 2{\mathrm{e}}^{\mathrm{x}} + \frac{{\mathrm{x}}^3}{3} + \mathrm{C}$

• ${\mathrm{e}}^{\mathrm{y}} = 2{\mathrm{e}}^{- \mathrm{x}} + \frac{{\mathrm{x}}^3}{3} + \mathrm{C}$

The solution of the differential equation $\left(\mathrm{x} +\hspace{0.17em}2{\mathrm{y}}^3\right)\frac{\mathrm{dy}}{\mathrm{dx}} = \mathrm{Y}$ is

• y³ + Cx = y

• $\frac{{\mathrm{xy}}^4}{2} + \mathrm{xy} = \mathrm{Cy}$

• y³ + Cy = x

• x + 2y³ = y + C

The solution ofthe differential equation

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

• x = ye^cy

• y = xe^cy

• x = ye^cx

• None of these

The solution of the differential equation$\frac{\mathrm{dy}}{\mathrm{dx}} + \sin \left(\frac{\mathrm{y} + \mathrm{x}}{2}\right) + \sin \left(\frac{\mathrm{y} - \mathrm{x}}{2}\right) = 0$ is

• $\log \left(\tan \left(\frac{\mathrm{y}}{2}\right)\right) = \mathrm{C} - 2\sin \left(\mathrm{x}\right)$

• $\log \left(\tan \left(\frac{\mathrm{y}}{4}\right)\right) = \mathrm{C} - 2\sin \left(\frac{\mathrm{x}}{2}\right)$

• $\log \left(\tan \left(\frac{\mathrm{y}}{2} + \frac{\mathrm{\pi }}{4}\right)\right) = \mathrm{C} - 2\sin \left(\mathrm{x}\right)$

• $\log \left(\tan \left(\frac{\mathrm{y}}{2} + \frac{\mathrm{\pi }}{4}\right)\right) = \mathrm{C} - 2\sin \left(\frac{\mathrm{x}}{2}\right)$

The solution of differential equation $\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right) - 2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} = 0$ is

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = \mathrm{xC}$

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

• x² + y² = C

• x² - y² = C

The solution of differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{x}\left(2\log \left(\mathrm{x}\right) + 1\right)}{\sin \left(\mathrm{y}\right) + \mathrm{ycos}\left(\mathrm{y}\right)}$ is

• $\mathrm{ysin}\left(\mathrm{y}\right) = {\mathrm{x}}^2\log \left(\mathrm{x}\right) + \mathrm{C}$

• $\mathrm{y} = {\mathrm{x}}^2 + \log \left(\mathrm{x}\right) + \mathrm{C}$

• $\mathrm{ysin}\left(\mathrm{y}\right) = {\mathrm{x}}^2 + \mathrm{C}$

• None of these

557.

Solution of $2\mathrm{ysin}\left(\mathrm{x}\right)\frac{\mathrm{dy}}{\mathrm{dx}} = 2\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right) - {\mathrm{y}}^2\cos \left(\mathrm{x}\right), \mathrm{x} = \frac{\mathrm{\pi }}{2}$, y = 1 is given by

• y² = sin(x)

• y = sin²(x)

• y² = cos(x) + 1

• None of these

Solution of ${\mathrm{x}}^2\frac{\mathrm{dy}}{\mathrm{dx}} - \mathrm{xy} = 1 +\hspace{0.17em}\cos \left(\frac{\mathrm{y}}{\mathrm{x}}\right)$ is

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

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

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

• ${\mathrm{x}}^2 = \left(\mathrm{C} + {\mathrm{x}}^2\right)\tan \left(\raisebox{1ex}{$\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.\right)$

The solution of $\frac{\mathrm{dy}}{\mathrm{dx}} = \cos \left(\mathrm{x}\right)\left(2 - \mathrm{ycsc}\left(\mathrm{x}\right)\right)$ where y = 2, when x = $\frac{\mathrm{\pi }}{2}$ is

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

• $\mathrm{y} = \tan \left(\frac{\mathrm{x}}{2}\right) + \cot \left(\frac{\mathrm{x}}{2}\right)$

• $\mathrm{y} = \frac{1}{\sqrt{2}}\sec \left(\frac{\mathrm{x}}{2}\right) + \sqrt{2}\cos \left(\frac{\mathrm{x}}{2}\right)$

• None of the above

The solution of the equation ${\sin }^{-1}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = \mathrm{x} + \mathrm{y}$ is

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

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

• $\tan \left(\mathrm{x} + \mathrm{y}\right) - \sec \left(\mathrm{x} + \mathrm{y}\right) + \mathrm{x} + \mathrm{C} = 0$

• None of the above