Find all real valued $C^1$ solutions for the equation $xy'(x)+y(x)=x$, for $x \in (-1,1)$.
My query is: can I put $x=0$ in the general solution of that differential equation, as $x=0$ is well within the given range $(-1,1)$ for $x$?
I found a general solution of: $2yx=x^2+c$. If I put $x=0$ we will obtain a solution of $y=x/2$.
The given question asks for all real-valued $C^1$ solutions.
\begin{align} x y' + y&=x \\ \implies (xy)' &= x\\ \implies xy &= \frac{1}{2}x^2 + C \\ \implies y&=\frac{1}{2}x+\frac{C}{x} \end{align} So please be aware of letting $x=0$.In fact I do not like your range beign set as $x \in (-1, +1)$ either.