Help with solve of Differential Equation from lines of curvature

Question

I was resolving a problem from Do Carmo, where I need to calculate the curvature lines of a helicoid. By the last part, what I got is: $$ u'+ \sqrt{v^2 + c^2} = v'. $$ I've tried to integrate in relation to $u$ first and got this: $$ u\sqrt{v^2 + c^2} - M_{1} = uv'. $$ When I integrate in $v$, this appears: $$ u\left(\frac{1}{2}v\sqrt{v^2+c^2} + \frac{c^2}{2}\log{\left(\frac{\left|v + \sqrt{c^2 + v^2} \right|}{|c|}\right)} + M_{2}\right) - \lambda v + M_{3} = vu + M_{4}, $$ where $M_{i}$ are constants. How to prove that this equation is equivalent to $$ \log{\left(v + \sqrt{v^2 + c^2} \right)} \pm u = \pm M, $$ where $M$ is a constant? Thanks in advance!!