261.

The general solution of the differential equation $\sqrt{1 - {\mathrm{x}}^2{\mathrm{y}}^2} . \mathrm{dx} = \mathrm{y} . \mathrm{dx} + \mathrm{x} . \mathrm{dy}$ is

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

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

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

• $\sin \left(\mathrm{xy}\right) + \mathrm{x} = \mathrm{C}$

If m and n are order and degree of the differential equation ${\left(\mathrm{y}\text{'}\text{'}\right)}^5 + \frac{{\left(\mathrm{y}\text{'}\text{'}\right)}^3}{\mathrm{y}\text{'}\text{'}\text{'}} + \mathrm{y}\text{'}\text{'}\text{'} = \sin \left(\mathrm{x}\right)$, then

• m = 3, n = 5

• m = 3, n = 1

• m = 3, n = 3

• m = 3, n = 2

The order and degree of the differential equation y = $\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{2}{{\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}}}$ is

• 1, 2

• 1, 3

• 2, 1

• 1, 1

The general solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{\mathrm{y}}{\mathrm{x}} = 3\mathrm{x}$ is

• $\mathrm{y} = \mathrm{x} - \frac{\mathrm{C}}{\mathrm{x}}$

• $\mathrm{y} = \mathrm{x} + \frac{\mathrm{C}}{\mathrm{x}}$

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

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

The order of differential equation of all circles of given radius 'a' is

• 1

• 4

• 3

• 2

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

• $\mathrm{y} = \frac{{\mathrm{x}}^4 + \mathrm{C}}{{\mathrm{x}}^2}$

• $\mathrm{y} = \frac{{\mathrm{x}}^2 + \mathrm{C}}{4{\mathrm{x}}^2}$

• $\mathrm{y} = \frac{{\mathrm{x}}^4 + \mathrm{C}}{4{\mathrm{x}}^2}$

• $\mathrm{y} = \frac{{\mathrm{x}}^2}{4} + \mathrm{C}$

The differential coefficient of log₁₀(x) with respect to log_x(10) is

• 1

• $- {\left({\log }_{10}\left(\mathrm{x}\right)\right)}^2$

• ${\left({\log }_{\mathrm{x}}\left(10\right)\right)}^2$

• $\frac{{\mathrm{x}}^2}{100}$

The solution for the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{\mathrm{dx}}{\mathrm{x}} = 0$ is

• $\frac{1}{\mathrm{y}} + \frac{1}{\mathrm{x}} = \mathrm{C}$

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

• xy = C

• x + y = C

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

• order = 2, degree = 3

• order = 2, degree = 4

• oreder = 2, degree = $\frac{3}{4}$

• order = 2, degree = not defined

Integrating factor of x$\frac{\mathrm{dy}}{\mathrm{dx}} - \mathrm{y} = {\mathrm{x}}^4 - 3\mathrm{x}$ is

• x

• log(x)

• $\frac{1}{\mathrm{x}}$

• - x