31.

The solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{2\mathrm{yx}}{1 +\hspace{0.17em}{\mathrm{x}}^2} = \frac{1}{{\displaystyle {\left(1 +\hspace{0.17em}{\mathrm{x}}^2\right)}^2}}$ is :

• y(1 + x²) = c + tan^-1(x)

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

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

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

The solution of the differential equation $\mathrm{xdy} - \mathrm{ydx} = \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2}\mathrm{dx}$ is :

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

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

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

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

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

• y = e^{x - y} + x²e^{- y} + c

• ${\mathrm{e}}^{\mathrm{y}} - {\mathrm{e}}^{\mathrm{x}} = \frac{1}{3}{\mathrm{x}}^3 + \mathrm{c}$

• ${\mathrm{e}}^{\mathrm{y}} + {\mathrm{e}}^{\mathrm{x}} = \frac{1}{3}{\mathrm{x}}^3 + \mathrm{c}$

• ${\mathrm{e}}^x - {\mathrm{e}}^y = \frac{1}{3}{\mathrm{x}}^3 + \mathrm{c}$