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

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

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

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

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

The integrating factor of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{\mathrm{y}}{\mathrm{x}} = {\mathrm{x}}^3 - 3$ will be

• x

• log(x)

• - x

• e^x

The solution of xdx + ydy = x²ydy - xy²dx is

• x² - 1 = C(1 + y²)

• x² + 1 = C(1 - y²)

• x² - 1 = C(1 - y²)

• x² + 1 = C(1 - y²)

The solution of x² + y²$\frac{\mathrm{dy}}{\mathrm{dx}}$ = 4 is

• x² + y² = 12x + C

• x² + y² = 3x + C

• x² + y² = 8x + C

• x³ + y³ = 12x + C

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

• 2y = e^2x + C

• 2ye^x = e^x + C

• 2ye^x = e^2x + C

• 2ye^2x = 2e^x + C

Order of the differential equation of the family of all concentric circles centered at (h, k) is

• 1

• 2

• 3

• 5

The solution of $\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{1}{3}\mathrm{y} = 1 \mathrm{is}$

• $\mathrm{y} = 3 + {\mathrm{ce}}^{\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

• $\mathrm{y} = 3 + {\mathrm{ce}}^{\raisebox{1ex}{$ - \mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

• $3\mathrm{y} = \mathrm{c} + {\mathrm{e}}^{\raisebox{1ex}{$ \mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

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

$\mathrm{y} + {\mathrm{x}}^2 =\frac{\mathrm{dy}}{\mathrm{dx}}$ has the solution

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

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

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

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

$\mathrm{The} \mathrm{solution} \mathrm{of} \frac{\mathrm{dy}}{\mathrm{dx}} = {\left(\frac{\mathrm{x}}{\mathrm{y}}\right)}^{ - \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} \mathrm{is}$

• ${\mathrm{x}}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} + {\mathrm{y}}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} = \mathrm{c}$

• ${\mathrm{y}}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} - {\mathrm{x}}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} = \mathrm{c}$

• ${\mathrm{x}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} + {\mathrm{y}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} = \mathrm{c}$

• ${\mathrm{y}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} - {\mathrm{x}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} = \mathrm{c}$

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

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

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

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

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