I am asked to find the general solution of the following ODE, as well as to give the largest interval of definition and determine if there are any transient terms in the general solution.
$$x\frac{dy}{dx} + (3x+1)y = e^{-3x}$$ integratingfactorde
As far as I know, pic-related is the complete solution. The book's solution insists I: (0,∞).
My question is why can't the interval be (-∞,0), because of the ln|x|? Why do so many methods seem keen on dropping the abs(...) from the solution and give "just because" as an explanation? Is there a mathematical reason for any of this or is dropping the abs(...) and setting the interval(s) to positive numbers just to make life easier?
You are right that you can solve this ODE both on $(-∞,0)$ and on $(0,∞)$. Which one to choose depends on the initial value, if that is given for a negative $x$, then $ (-∞,0)$ is the domain of the solution. If one interval has to be selected, the convention for the default choice is the positive interval.
You can write down the solution more compactly if you use the integrating factor to compress the left side in the other direction, as done in the copied picture $$ \frac{d}{dx}(xe^{3x}y(x))=xe^{3x}y'(x)+(3x+1)e^{3x}y(x)=1 $$ so that $$ xe^{3x}y(x) = x+C. $$ One should avoid the division by $x$ if in the next step a multiplication by $x$ cancels it out.