# Removing absolute value signs when solving differential equations and constant solutions

When solving the differential equation $$y' = 1-y^2$$ you get the solution $$|\frac{y+1}{y-1}| = Ce^{2x}$$ You can then remove the absolute value sign by changing C to a new konstant $K = \pm C$. But why is this? I've been struggling really hard to grasp this concept, and I'm also finding it hard to have an intuitive understanding of what the absolute value sign actually means practically in this context. What would would be the difference between having the absolute value sign surrounding our fraction and it not being there?

Also, I've been told the the same differential equation also has the two constant solution $K = \pm 1$. From what i understand constant solutions are found by setting $Y = K$, but what do they actually mean, and what do you do if there is an x in the equation?

• write $$e^{2x+C}=e^{2x}\cdot e^{C}$$ and set $$e^{C}=C'$$ – Dr. Sonnhard Graubner Nov 29 '17 at 18:09
• I don't see how that helps remove the absolute value sign. – Pame Nov 29 '17 at 18:25
• you can write $$(-1)C'=C''$$ – Dr. Sonnhard Graubner Nov 29 '17 at 18:26
• Yes, thats how you go about removing the absolute value sign like i described in my post. The problem is i don't see why it works. – Pame Nov 29 '17 at 18:53