281.

The degree and order of the differential equation of the family of all parabolas whose axis is X-axis, are respectively

• 2, 1

• 1, 2

• 3, 2

• 2, 3

Solution of the differential equation ydx - xdy = x²ydx

• ${\mathrm{ye}}^{{\mathrm{x}}^2} = {\mathrm{cx}}^2$

• ${\mathrm{ye}}^{- {\mathrm{x}}^2} = {\mathrm{cx}}^2$

• ${\mathrm{y}}^2{\mathrm{e}}^{{\mathrm{x}}^2} = {\mathrm{cx}}^2$

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

The solution of the differential equation $\left(1 + {\mathrm{y}}^2\right) + \left(\mathrm{x} - {\mathrm{e}}^{{\tan }^{-1}\left(\mathrm{y}\right)}\right)\frac{d\mathrm{y}}{d\mathrm{x}} = 0$ is

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

• $2{\mathrm{xe}}^{{\tan }^{-1}\left(\mathrm{y}\right)} = {\mathrm{e}}^{2{\tan }^{-1}\left(\mathrm{y}\right)} + \mathrm{c}$

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

• ${\mathrm{xe}}^{2{\tan }^{-1}\left(\mathrm{y}\right)} = {\mathrm{e}}^{{\tan }^{-1}\left(\mathrm{y}\right)} + \mathrm{c}$

Solution of the differential equation (x + y - 1)dx + (2x+ 2y - 3)dy = 0

• y + x + log (x + y - 2) = c

• y + 2x + log (x + y - 2) = c

• 2y + x + log (x + y - 2) = c

• 2y + 2x + log (x + y - 2) = c

The differential equation for the family of curve x² + y² - 2ay = 0, where a is an arbitrary constant, is

• 2(x² - y²)y' = xy

• 2(x² + y²)y' = xy

• (x² - y²)y' = 2xy

• (x² + y²)y' = 2xy

Solution of the differential equation $\cos \left(\mathrm{x}\right)\frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{ysin}\left(\mathrm{x}\right) = 1$ is

• $\mathrm{ysec}\left(\mathrm{x}\right) + \tan \left(\mathrm{x}\right) = \mathrm{c}$

• $\mathrm{ysec}\left(\mathrm{x}\right) = \tan \left(\mathrm{x}\right) + \mathrm{c}$

• $\mathrm{ytan}\left(\mathrm{x}\right) = \sec \left(\mathrm{x}\right) + \mathrm{c}$

• $\mathrm{ytan}\left(\mathrm{x}\right) = \sec \left(\mathrm{x}\right)\tan \left(\mathrm{x}\right) + \mathrm{c}$

The solution of the differential equation ydx + (x + x²y)dy = 0 is

• $\frac{- 1}{\mathrm{xy}} = \mathrm{c}$

• $\frac{- 1}{\mathrm{xy}} + \log \left(\mathrm{y}\right) = \mathrm{c}$

• $\frac{1}{\mathrm{xy}} = \log \left(\mathrm{y}\right) + \mathrm{c}$

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

The differential equation of the equation y² = m(x² - a²) is

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

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

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

• None of the above

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{ax} +\hspace{0.17em}\mathrm{h}}{\mathrm{by} + \mathrm{k}}$ will be parabola, if

• a = 0

• b = 1

• a = 1

• None of these

The number of real solutions of ${\tan }^{-1}\left(\sqrt{\mathrm{x}\left(\mathrm{x} + 1\right)}\right) + {\sin }^{-1}\left(\sqrt{{\mathrm{x}}^2 + \mathrm{x} + 1}\right) = \frac{\mathrm{\pi }}{2}$ is

• zero

• one

• two

• infinite