Solve each of the following initial value problems: `x(dy)/(dx)+y=xlogx ,y(1)=1/4`

Explanation

$$
\begin{aligned}
&x \frac{d y}{d x}-y-\log x y(1)=0 \\
&\text { Given } x \frac{y}{d x}-y-\log x \text { and } y(1)=0 \\
&\rightarrow\left(x \frac{d y}{d x}-y\right) x \frac{1}{x}-\log x x \frac{1}{x} \\
&\Rightarrow \frac{d y}{d x}-\frac{y}{x}=\frac{\log x}{x} \\
&\rightarrow \frac{d y}{d x}+\left(\frac{-1}{x}\right) y=\frac{\log x}{x}
\end{aligned}
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