Given the initial value problem of Euler Differential Equation: $$x^2y''+\beta xy'+\alpha y=0$$ $$y(-1)=2 , y'(-1)=3$$
According to my book, the general solution for x<0 is the same as that of x > 0 so all the possible general solutions should be expressed with absolute value of x. For example, if we assume $y= x^r$ then if the characteristic equation resulted in a repeated root for r, then the general solution would be $y= c_1\left | x \right |^r+ c_2\left | x \right |^rln(\left | x \right |)$ with the absolute value for both x terms. However, when I encountered an example where the initial conditions were at a negative x point, my book expressed the general solution without the absolute value of x which gave a different answer from the one with the general solution expressed with the absolute value of x. So which is correct, expresseing the general solution with or without the absolute values?