681.

The differential equation of the family of parabola with focus as the origin and the axis as X-axis, is

• $\mathrm{y}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 + 4\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} = 4\mathrm{y}$

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

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

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

Soution of $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{xlog}\left({\mathrm{x}}^2\right) + \mathrm{x}}{\sin \left(\mathrm{y}\right){\displaystyle }{\displaystyle +}{\displaystyle }{\displaystyle \mathrm{ycos}}{\displaystyle \left(\mathrm{y}\right)}}$ is

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

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

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

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

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

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

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

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

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

Integrating factor of (x + 2y³)$\frac{\mathrm{dy}}{\mathrm{dx}}$ = y² is

• ${\mathrm{e}}^{\left(\frac{1}{\mathrm{y}}\right)}$

• ${\mathrm{e}}^{- \left(\frac{1}{\mathrm{y}}\right)}$

• y

• $\frac{- 1}{\mathrm{y}}$

y = Ae^x + Be^2x + Ce^3x satisfies the differential equation

• y''' - 6y'' + 11y' - 6y = 0

• y''' + 6y'' + 11y' + 6y = 0

• y''' + 6y'' - 11y' + 6y = 0

• y''' - 6y'' - 11y' + 6y = 0

Observe the following statements

A. Integrating factor of $\frac{{\displaystyle \mathrm{dy}}}{{\displaystyle \mathrm{dx}}} + \mathrm{y} = {\mathrm{x}}^2$ is e^x

R. Integrating factor of $\frac{{\displaystyle \mathrm{dy}}}{{\displaystyle \mathrm{dx}}} + \mathrm{P}\left(\mathrm{x}\right)\mathrm{y} = \mathrm{Q}\left(\mathrm{x}\right)$ is ${\mathrm{e}}^{\int \mathrm{P}\left(\mathrm{x}\right)\mathrm{dx}}$

• A is true, R is false

• A is false, R is true

• A is true, R is true, R $\Rightarrow$ A

• Both are false

$\mathrm{If} \mathrm{dx} + \mathrm{dy} = (\mathrm{x} + \mathrm{y})\left(\mathrm{dx} - \mathrm{dy}\right), \mathrm{then} \log \left(\mathrm{x} + \mathrm{y}\right) = ?$

• x + y + c

• x + 2y + c

• x - y + c

• 2x + y + c

$\mathrm{If} {\mathrm{x}}^2\mathrm{y} - {\mathrm{x}}^3\frac{\mathrm{dy}}{\mathrm{dx}} = {\mathrm{y}}^4\mathrm{cosx}, \mathrm{then} {\mathrm{x}}^3{\mathrm{y}}^{- 3} \mathrm{is} \mathrm{equal} \mathrm{to}$

• sin(x)

• 2sin(x) + c

• - 3sin(x) + c

• 3cos(x) + c

Observe the following statements

$$\begin{gathered} \mathrm{I}. \mathrm{If} \mathrm{dy} +\hspace{0.17em}2\mathrm{xydx} = 2{\mathrm{e}}^{- {\mathrm{x}}^2}\mathrm{dx}, \mathrm{then} {\mathrm{ye}}^{{\mathrm{x}}^2} = 2\mathrm{x} + \mathrm{c}, \\ \mathrm{II}. \mathrm{If} {\mathrm{ye}}^{- {\mathrm{x}}^2} - 2\mathrm{x} = \mathrm{c}, \mathrm{then} \\ \mathrm{dx} = \left(2{\mathrm{e}}^{- {\mathrm{x}}^2} - 2\mathrm{xy}\right)\mathrm{dy} \\ \mathrm{which} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} \mathrm{statement} \end{gathered}$$

• Both I and II are true

• Neither I nor II is true

• I is false, But II is true

• I is true, But II is false

$\mathrm{If} \frac{\mathrm{dy}}{\mathrm{dx}} =\frac{ \mathrm{y} +\hspace{0.17em}\mathrm{xtan}\left({\displaystyle \frac{\mathrm{y}}{\mathrm{x}}}\right)}{\mathrm{x}}, \mathrm{then} \sin \left(\frac{\mathrm{y}}{\mathrm{x}}\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• cx²

• cx

• cx³

• cx⁴