I'm currently studying ODE's using Advanced Engineering Mathematics 10e (Kreyszig) and had a question regarding solutions in square root form. Specifically, the exercise problem is:
Find a general solution for the following ODE(s).
$yy' + 36x = 0$
My approach is as follows:
$$ yy' + 36x = 0 $$
$$ \begin{align} ydy & = -36xdx \\\ \frac{1}{2}y^2 & = -18x^2 + C \\\ y^2 & = -36x^2 + C \\\ y & = \pm \sqrt{-36x^2 + C} \\\ \end{align} $$
However, I noticed that in the solution, the author omits the $\pm$. Is there a reason why it has been omitted? Further, is there a rule-of-thumb that I should keep in mind regarding omission? Thank you.
The function $y=-\sqrt{-36x^2+C}$ is also a solution to the equation (you can check this).
Either omitting the $\pm$ is an error in the book, or there's some more information about the problem you're not telling us that would allow us to conclude that $y$ is positive in this case.