(a) Find the general solution of the differential equation: d/(d x)( x y^2)=2 y (1+ x^2)

Given differential equation is

$$\begin{aligned} & $\displaystyle 2$ \mathrm{xy} \frac{d \mathrm{y}}{d \mathrm{x}}+\mathrm{y}^{$\displaystyle 2$}=$\displaystyle 2$ \mathrm{y}\left($\displaystyle 1$+\mathrm{x}^{$\displaystyle 2$}\right) & \Rightarrow \frac{d \mathrm{y}}{d \mathrm{x}}+\frac{\mathrm{y}}{$\displaystyle 2$ \mathrm{x}}=\frac{1}{\mathrm{x}}+\mathrm{x} \end{aligned}

$$Integrating factor $\displaystyle =\mathrm{e}^{\int \frac{1}{2 \mathrm{x}} d \mathrm{x}}=\mathrm{e}^{\log \sqrt{\mathrm{x}}}=\sqrt{\mathrm{x}}$

Solution is given by $\displaystyle \mathrm{y} \sqrt{\mathrm{x}}=\int\left(\frac{1}{\sqrt{\mathrm{x}}}+\mathrm{x}^{\frac{3}{2}}\right) \mathrm{dx}$

$$\Rightarrow \mathrm{y} \sqrt{\mathrm{x}}=$\displaystyle 2$ \sqrt{\mathrm{x}}+\frac{$\displaystyle 2$ \mathrm{x}^{\frac{5}{2}}}{$\displaystyle 5$}+\mathrm{C} \text {, or } \mathrm{y}=$\displaystyle 2$+\frac{$\displaystyle 2$ \mathrm{x}^{$\displaystyle 2$}}{$\displaystyle 5$}+\frac{\mathrm{C}}{\sqrt{\mathrm{x}}}

$$(b) Given differential equation is $\displaystyle \frac{d \mathrm{y}}{d \mathrm{x}}=\frac{\mathrm{y}}{\mathrm{x}}-\mathrm{e}^{\frac{\mathrm{y}}{\mathrm{x}}}$

Let $\displaystyle \mathrm{y}=\mathrm{vx} \Rightarrow \frac{d \mathrm{y}}{d \mathrm{x}}=\mathrm{v}+\mathrm{x} \frac{d \mathrm{v}}{d \mathrm{x}}$

The given equation becomes $\displaystyle \mathrm{v}+\mathrm{x} \frac{d \mathrm{v}}{d \mathrm{x}}=\mathrm{v}-\mathrm{e}^{\mathrm{v}}$

$$\Rightarrow-\mathrm{e}^{-\mathrm{v}} \mathrm{dv}=\frac{d \mathrm{x}}{\mathrm{x}}

$$Integrating both sides, we get

$$\begin{aligned} & \mathrm{e}^{-\mathrm{V}}=\log |\mathrm{x}|+\mathrm{C} & \Rightarrow \mathrm{e}^{-\frac{\mathrm{y}}{\mathrm{x}}}=\log |\mathrm{x}|+\mathrm{C} \end{aligned}