22.

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

$$\begin{gathered} \mathrm{Given}, \left(\mathrm{x} +\hspace{0.17em}2{\mathrm{y}}^3\right)\frac{\mathrm{dy}}{\mathrm{dx}} = \mathrm{y} \\ \mathrm{y}\frac{\mathrm{dx}}{\mathrm{dy}} = \mathrm{x} + 2{\mathrm{y}}^3 \\ \Rightarrow \frac{\mathrm{dx}}{\mathrm{dy}} - \frac{1}{\mathrm{y}}\mathrm{x} = 2{\mathrm{y}}^2 \\ \mathrm{This} \mathrm{is} \mathrm{the} \mathrm{form} \mathrm{of} \frac{\mathrm{dx}}{\mathrm{dy}} + \mathrm{Px} = \mathrm{Q} \\ \mathrm{where}, \mathrm{P} = - \frac{1}{\mathrm{y}} \mathrm{and} \mathrm{Q} = 2{\mathrm{y}}^2 \\ \mathrm{Thus}, \mathrm{the} \mathrm{given} \mathrm{equation} \mathrm{is} \mathrm{linear}. \end{gathered}$$

$\therefore \mathrm{IF} = {\mathrm{e}}^{\int \mathrm{Pdy}} = {\mathrm{e}}^{\int - \frac{1}{\mathrm{y}}\mathrm{dy}} = {\mathrm{e}}^{- \log \left(\mathrm{y}\right)} = {\mathrm{e}}^{\log {\left(\mathrm{y}\right)}^{- 1}} = {\mathrm{y}}^{ - 1} = \frac{1}{\mathrm{y}}$

$$\begin{gathered} \mathrm{So}, \mathrm{the} \mathrm{required} \mathrm{solution} \mathrm{is} \\ \mathrm{x} . \mathrm{IF} = \int \left(\mathrm{Q} . \mathrm{IF}\right)\mathrm{dy} + \mathrm{C} \\ \Rightarrow \mathrm{x} . \frac{1}{\mathrm{y}} = \int \left(2{\mathrm{y}}^2 . \frac{1}{\mathrm{y}}\right)\mathrm{dy} + \mathrm{C} \\ \Rightarrow \mathrm{x} . \frac{1}{\mathrm{y}} = \int 2\mathrm{ydy} + \mathrm{C} = {\mathrm{y}}^2 + \mathrm{C} \\ \Rightarrow \mathrm{x} = {\mathrm{y}}^3 + \mathrm{Cy} \end{gathered}$$