Question: Identify each of the differential equations in Problems 1to 24 as to type (for example, separable, linear first order, linear second order, etc.), and then solve it.

${\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathit{x}\mathit{y}\mathbf{=}\mathit{x}\mathbf{/}\mathit{y}$

The differential equation is$y^{\text{'}}+xy=\frac{x}{y}.$

When fand its derivatives are inserted into the equation, a solution is a function y=f(x) that solves the differential equation. The highest order of any derivative of the unknown function appearing in the equation is the order of a differential equation.

A differential equation of the form$\mathbf{(}\mathit{D}\mathbf{-}\mathit{a}\mathbf{)}\mathbf{(}\mathit{D}\mathbf{-}\mathit{b}\mathbf{)}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{a}\mathbf{\ne }\mathit{b}$ has general solution$\mathit{y}\mathbf{=}{\mathit{c}}_{\mathbf{1}}{\mathit{e}}^{\mathbf{a}\mathbf{x}}\mathbf{+}{\mathit{c}}_{\mathbf{2}}{\mathit{e}}^{\mathbf{b}\mathbf{x}}\mathbf{.}$

Consider the equation.

$y^{\text{'}}+xy=\frac{x}{y}$

Rearrange above equation.

role="math" localid="1664363476430" $$\begin{gathered} \begin{array}{c}\frac{dy}{dx}+xy=\frac{x}{y}\\ \frac{dy}{dx}=\frac{x}{y}-xy\\ \frac{ydy}{1-y}=xdx\\ \frac{-ydy}{1-y}=-xdx\end{array} \\ \left(1-\frac{1}{1-y}\right)dy=-xdx \end{gathered}$$

The above equation it is clear that the equation can be solved by separation of variables.

Integrate above equation.

$\begin{array}{c}\int \left(1-\frac{1}{1-y}\right)dy=\int -xdx\\ y+\ln (1-y)=-\frac{x^2}{2}+C\\ 2y+2\ln (1-y)+x^2=2C\end{array}$

Thus, the solution of given differential equation is $2y+2\ln (1-y)+x^2=2C$.