The solution of the equation $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{\mathrm{x} + \mathrm{y} + 1}$ is

• x + y = Ce^y - 2

• x + y = Clog(y) - 4

• log(x+ y + 2) = Cy

• log(x + y + 2) = C + y

The solution of differential equation (ylog(x) - 1)ydx = xdy is

• $\mathrm{y}\left(\log \left({\mathrm{e}}^{\mathrm{x}}\right) + \mathrm{Cx}\right) = 1$

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

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

• None of these

The solution of the differential equation $\sqrt{\mathrm{a} + \mathrm{x}}\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{xy} = 0$ is

• $\mathrm{y} = {\mathrm{Ce}}^{\frac{2}{3}\left(2\mathrm{a} - \mathrm{x}\right)\sqrt{\mathrm{x} + \mathrm{a}}}$

• $\mathrm{y} = {\mathrm{Ce}}^{\frac{2}{3}\left(\mathrm{a} - \mathrm{x}\right)\sqrt{\mathrm{x} + \mathrm{a}}}$

• $\mathrm{y} = {\mathrm{Ce}}^{\frac{2}{3}\left(2\mathrm{a} + \mathrm{x}\right)\sqrt{\mathrm{x} + \mathrm{a}}}$

• $\mathrm{y} = {\mathrm{Ce}}^{- \frac{2}{3}\left(2\mathrm{a} - \mathrm{x}\right)\sqrt{\mathrm{x} + \mathrm{a}}}$

The general solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = \mathrm{ytan}\left(\mathrm{x}\right) - {\mathrm{y}}^2\sec \left(\mathrm{x}\right)$ is

• $\tan \left(\mathrm{x}\right) = \left(\mathrm{C} + \sec \left(\mathrm{x}\right)\right)\mathrm{y}$

• $\sec \left(\mathrm{y}\right) = \left(\mathrm{C} + \tan \left(\mathrm{y}\right)\right)\mathrm{x}$

• $\sec \left(\mathrm{x}\right) = \left(\mathrm{C} + \tan \left(\mathrm{x}\right)\right)\mathrm{y}$

• $\tan \left(\mathrm{y}\right) = \left(\mathrm{C} + \sec \left(\mathrm{x}\right)\right)\mathrm{x}$

The degree of the differential equation satisfying $\sqrt{1 - {\mathrm{x}}^2} + \sqrt{1 - {\mathrm{y}}^2} = \mathrm{a}\left(\mathrm{x} - \mathrm{y}\right)$ is

• 1

• 2

• 3

• None of the above

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

• y = C(x + a)(1 - ay)

• y = C(x + a)(a + ay)

• y = C(x - a)(1 - ay)

• None of the above

The solution of differential equation (2y - 1)dx - (2x + 3)dy = 0 will be

• $\frac{2\mathrm{x} - 1}{2\mathrm{y} + 3} = \mathrm{C}$

• $\frac{2\mathrm{y} + 1}{2\mathrm{x} - 3} = \mathrm{C}$

• $\frac{2\mathrm{x} + 3}{2\mathrm{y} - 1} = \mathrm{C}$

• $\frac{2\mathrm{x} - 1}{2\mathrm{y} - 1} = \mathrm{C}$

The solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{xlog}\left({\mathrm{x}}^2\right) + \mathrm{x}}{\sin \left(\mathrm{y}\right){\displaystyle }{\displaystyle +}{\displaystyle }{\displaystyle \mathrm{ycos}}{\displaystyle \left(\mathrm{y}\right)}}$ will be

• ysin(y) = x²log(x) + C

• ysin(y) = x² + C

• ysin(y) = x² + lo(x) + C

• ysin(y) = xlog(x) + C

The order and degree of the differential equation ${\left(\mathrm{y}\text{'}\right)}^4 = \sqrt{1 - \mathrm{y}\text{'}\text{'}\text{'}}$ are respectively.

• 3, 4

• 1, 2

• 3, 2

• 3, 1

The differential equation of the family of circles having centre on Y-axis and radius 4 is

• $\left({\mathrm{x}}^2 - 4\right){\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 + {\mathrm{x}}^2 = 0$

• $\left({\mathrm{x}}^2 - 9\right){\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 + {\mathrm{x}}^2 = 0$

• $\left({\mathrm{x}}^2 - 9\right)\frac{\mathrm{dy}}{\mathrm{dx}} + {\mathrm{x}}^2 = 0$

• $\left({\mathrm{x}}^2 - 16\right){\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 + {\mathrm{x}}^2 = 0$