# Eliminating Absolute Value in the solution of a Differential Equation

solving this simple Cauchy problem $$y' = ry$$ with initial condition $$y(0) = b$$, the general integral is $$ln|y|=rx+c$$ or $$|y|=e^{rx+c}=e^{c}e^{rx}= ke^{rx}$$.

My question is: what are the steps to eliminate the absolute value from the solution?

• Note that $-y$ solves $y'=ry$ whenever $y$ solves it. – Christoph Nov 2 '18 at 17:45