Suppose I am given a Cauchy-Euler form second order differential equation

${x}^{2}\frac{{d}^{2}y}{d{x}^{2}}+x\frac{dy}{dx}+y=f(x).$

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 $x={e}^{t}$. But I have a fundamental doubt here, we know that ${e}^{t}>0$ $\mathrm{\forall}t\in \mathbb{R}$. But when we are using the above transformation we are subconsciously assuming $x>0$. How does this make sense?

Substituting $t=\mathrm{ln}(x)$ also makes no difference as ln is defined on ${\mathbb{R}}_{>0}$. 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?

${x}^{2}\frac{{d}^{2}y}{d{x}^{2}}+x\frac{dy}{dx}+y=f(x).$

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 $x={e}^{t}$. But I have a fundamental doubt here, we know that ${e}^{t}>0$ $\mathrm{\forall}t\in \mathbb{R}$. But when we are using the above transformation we are subconsciously assuming $x>0$. How does this make sense?

Substituting $t=\mathrm{ln}(x)$ also makes no difference as ln is defined on ${\mathbb{R}}_{>0}$. 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?