Differential equations and the consequence of its scale invariance This is a question based on a Wikipedia article on scale invariance (see the claim under the section Classical Electromagnetism).
The equation $$\frac{\partial^2\psi(x,t)}{\partial x^2}=\frac{\partial^2\psi(x,t)}{\partial t^2}\tag{1}$$ is invariant under $x,t\rightarrow \lambda x,\lambda t$. Wikipedia claims that if $\psi(x,t)$ is a solution, $\psi(\lambda x,\lambda t)$ is also a solution.


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*Is this claim independent of the boundary condition?

*All solutions of the type $\psi(\lambda x,\lambda t)$ obtained from a particular $\psi(x,t)$, form only a subset of all possible solutions of (1) i.e., they do not exhaust all possible solutions. Am I correct?

*

*Yes.

*Of course one can add particular solutions to any general solution, i.e. if $\psi(x,t)$ is solution so will be $\psi(x,t)+Kx$ for any constant $K$.

