271.

General solution of differential equations $\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{y} = 1\left(\mathrm{y} \ne 1\right)$ is

• $\log \left|\frac{1}{1 - \mathrm{y}}\right| = \mathrm{x} + \mathrm{C}$

• $\log \left|1 - \mathrm{y}\right| = \mathrm{x} + \mathrm{C}$

• $\log \left|1 + \mathrm{y}\right| = \mathrm{x} + \mathrm{C}$

• $\log \left|\frac{1}{1 - \mathrm{y}}\right| = - \mathrm{x} + \mathrm{C}$

The degree of the differential equation ${\left[1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right]}^2 = \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is

• 3

• 2

• 1

• 4

The integrating factor of the differential equation $\mathrm{x} . \frac{\mathrm{dy}}{\mathrm{dx}} + 2\mathrm{y} = {\mathrm{x}}^2 \mathrm{is} \left(\mathrm{x} \ne 0\right)$

• x

• $\log \left|\mathrm{x}\right|$

• x²

• ${\mathrm{e}}^{\log \left(\mathrm{x}\right)}$

The order of the differential equation

$\mathrm{y}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = \frac{\mathrm{x}}{\left({\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}}\right)} + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^3$ is

• 1

• 2

• 3

• 4

The general solution of the differential equation (x + y)dx + xdy = 0 is

• x² + y² = c

• 2x² - y²

• x² + 2xy = c

• y² + 2xy = c

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

• 1, 2/3

• 3, 1

• 3, 3

• 1, 2

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

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

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

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

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

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

• $\mathrm{y} = \frac{{\mathrm{e}}^{- 2\mathrm{x}}}{4}$

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

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

• $\mathrm{y} = \frac{{\mathrm{e}}^{- 2\mathrm{x}}}{4} + \mathrm{c} + \mathrm{d}$

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

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

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

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

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

Family y = Ax + A³ ofcurve is represented by the differential equation ofdegree

• 3

• 2

• 1

• None of these