221.

The order and power of differential equation

$\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + 7\frac{\mathrm{dy}}{\mathrm{dx}} + \int \mathrm{ydx} = \sin \left(\mathrm{x}\right)$ is

• 1, 3

• 3, 1

• 1, 2

• 2, 1

The solution of differential equation ${\mathrm{xcos}}^2\left(\mathrm{y}\right)\mathrm{dx} = {\mathrm{ycos}}^2\left(\mathrm{x}\right)\mathrm{dx}$ is

• $\mathrm{xtan}\left(\mathrm{x}\right) - \mathrm{ytan}\left(\mathrm{y}\right) - \log \left(\sec \left(\mathrm{x}\right)/\sec \left(\mathrm{y}\right)\right) = \mathrm{c}$

• $\mathrm{ytan}\left(\mathrm{x}\right) - \mathrm{xtan}\left(\mathrm{x}\right) - \log \left(\sec \left(\mathrm{x}\right).\sec \left(\mathrm{y}\right)\right) = \mathrm{c}$

• $\mathrm{xtan}\left(\mathrm{x}\right) - \mathrm{ytan}\left(\mathrm{y}\right) + \log \left(\sec \left(\mathrm{x}\right).\sec \left(\mathrm{y}\right)\right) = \mathrm{c}$

• None of the above

Differential equation of those circles which passes through origin and their centres lie on y-axis will be

• $\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} +\hspace{0.17em}2\mathrm{xy} = 0$

• $\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} =\hspace{0.17em}2\mathrm{xy}$

• $\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} =\hspace{0.17em}\mathrm{xy}$

• $\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} +\hspace{0.17em}\mathrm{xy} = 0$

The differential equation of all circles touching the axis of y at origin and centre on the x-axis is given by

• $\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} - {\mathrm{x}}^2 + {\mathrm{y}}^2 = 0$

• $2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} - {\mathrm{x}}^2 - {\mathrm{y}}^2 = 0$

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

• None of these

The solution of the differential equation

$\left({\mathrm{e}}^{- 2\sqrt{\mathrm{x}}} - \frac{\mathrm{y}}{\sqrt{\mathrm{x}}}\right)\frac{\mathrm{dx}}{\mathrm{dy}} = 1$ is given by

• ${\mathrm{e}}^{2\sqrt{\mathrm{x}}} = 2\sqrt{\mathrm{x}} + \mathrm{c}$

• ${\mathrm{ye}}^{- 2\sqrt{\mathrm{x}}} = \sqrt{\mathrm{x}} + \mathrm{c}$

• $\mathrm{y} = \sqrt{\mathrm{x}}$

• None of these

The solution of the differential equation

$\frac{\mathrm{dy}}{\mathrm{dx}} = \sqrt{\frac{1 - {\mathrm{y}}^2}{1 - {\mathrm{x}}^2}}$ is

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

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

• ${\sin }^{-1}\left(\mathrm{xy}\right) = 2$

• None of these

The solution of differential equation (1 + x)ydx + (1 - y)x dy = 0 is

• ${\log }_{\mathrm{e}}\left(\mathrm{xy}\right) + \mathrm{x} - \mathrm{y} = \mathrm{c}$

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

• ${\log }_{\mathrm{e}}\left(\frac{\mathrm{x}}{\mathrm{y}}\right) - \mathrm{x} + \mathrm{y} = \mathrm{c}$

• ${\log }_{\mathrm{e}}\left(\mathrm{xy}\right) - \mathrm{x} + \mathrm{y} = \mathrm{c}$

The differential equation of all circles which passes through the origin and whose centre lies on y-axis is

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

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

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

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

The general solution of y²dx + (x² - xy + y²)dy = 0 is

• ${\tan }^{-1}\left(\frac{\mathrm{x}}{\mathrm{y}}\right) + \log \left(\mathrm{y}\right) + \mathrm{c} = 0$

• $2{\tan }^{-1}\left(\frac{\mathrm{x}}{\mathrm{y}}\right) + \log \left(\mathrm{x}\right) + \mathrm{c} = 0$

• $\log \left(\mathrm{y} + \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2}\right) + \log \left(\mathrm{y}\right) + \mathrm{c} = 0$

• ${\sinh }^{-1}\left(\frac{\mathrm{x}}{\mathrm{y}}\right) + \log \left(\mathrm{y}\right) + \mathrm{c} = 0$

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

• $\mathrm{y} = \frac{1}{4}{\mathrm{e}}^{- 2\mathrm{x}} + \frac{\mathrm{cx}}{2} +\hspace{0.17em}\mathrm{d}$

• $\mathrm{y} = \frac{1}{4}{\mathrm{e}}^{- 2\mathrm{x}} + \mathrm{cx} +\hspace{0.17em}\mathrm{d}$

• $\mathrm{y} = \frac{1}{4}{\mathrm{e}}^{- 2\mathrm{x}} + {\mathrm{cx}}^2 +\hspace{0.17em}\mathrm{d}$

• $\mathrm{y} = \frac{1}{4}{\mathrm{e}}^{- 2\mathrm{x}} + {\mathrm{cx}}^3 +\hspace{0.17em}\mathrm{d}$