21.

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

• ${\mathrm{e}}^{\mathrm{y}} = 2{\mathrm{e}}^{\mathrm{x}} + \frac{{\mathrm{x}}^3}{3} + \mathrm{C}$

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

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

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

The solution of the differential equation $\left(\mathrm{x} +\hspace{0.17em}2{\mathrm{y}}^3\right)\frac{\mathrm{dy}}{\mathrm{dx}} = \mathrm{Y}$ is

• y³ + Cx = y

• $\frac{{\mathrm{xy}}^4}{2} + \mathrm{xy} = \mathrm{Cy}$

• y³ + Cy = x

• x + 2y³ = y + C