I'm currently studying ODE's with the textbook Advanced Engineering Mathematics (Kreyszig) and had a question regarding the absolute value within the natural logarithm of the integration of $1/x$.
I'm not particular good at calculus in general, but I understand the reason for why the absolute value is there in the first place (i.e. $\ln{|x|} + C$) but I've noticed that in many cases the author seems to include or omit it, without particular explanation.
In particular, the specific example problem I was looking at that caused this question is related to orthogonal trajectories:
If we have a a family of ellipses:
$$\frac{1}{2}x^2 + y^2 = C$$
the orthogonal trajectories are obtained as follows:
- Differentiating both sides w.r.t. $x$ yields $x + 2yy' = 0$, which in turn gives
$$ y' = f(x, y) = -\frac{x}{2y}$$
- Finding an ODE for the orthogonal trajectories $\tilde{y} = \tilde{y}(x)$ is
$$\tilde{y}' = -\frac{1}{f(x, \tilde{y})} = \frac{2\tilde{y}}{x}$$
- Solving this ODE via separating variables, integrating, and taking exponents gives:
$$\ln(|\tilde{y}|) = 2\ln(x) + C,\quad \tilde{y} = Cx^2$$
As you can see, when separating variables and integrating, I noticed that the $\tilde{y}$'s absolute value is taken whereas $x$ is not. Also, the final general solution is $\tilde{y}$ rather than $|\tilde{y}|$.
What is the reasoning for this? Is there anything that I should keep in mind in general when dealing with the absolute values within the logarithms? Any help or tips are appreciated, as I feel I'll be needing them throughout the textbook.