The usual textbook method for solving the Cauchy Euler equation is to blackuce it to a linear differential equation with constant coefficients by the transformation . But I have a fundamental doubt here, we know that . But when we are using the above transformation we are subconsciously assuming . How does this make sense?
Substituting also makes no difference as ln is defined on . So I now doubt the validity of the method followed to solve the Cauchy-Euler equation.
Can someone give me a proper explanation of what exactly going on here and why the process is valid?