What does the adjective equidimensional mean ? I encountered it while studying Cauchy-euler equation in differential equations with applications and historical notes by gf simmons : http://matematicaeducativa.com/foro/download/file.php?id=2247&sid=f867258ad39908e15b331d33e763a8b0
Kindly see page no. 126, q5
Wikipedia says :
"A topological space X is said to be equidimensional if for all points p in X the dimension at p that is, dim p(X) is constant. The Euclidean space is an example of an equidimensional space."
But isn't this always the case and how is an ODE related with dimensions of a point ?
I am a first year undergraduate , please explain in simple or heuristic way.
A topological space is called equi-dimensional, if every irreducible component of X has the same dimension - see here. This is not always the case. The disjoint union of two spaces $X$ and $Y$ (as topological spaces) of different dimension is an example of a non-equidimensional space.
Edit: Now there is a link in the question, which shows that we talk about Euler's equidimensional equation. So what is the meaning of "equidimensional" for this equation?
Say for example the variable $y$ is a distance measured in meters $(\mbox{m})$ and the variable $x$ is time, measured in seconds $(\mbox{s})$. Then $y'$ is a velocity $(\mbox{m/s})$ and $y''$ is an acceleration $(\mbox{m/s}^2)$. Thus $x^2 y''$ is a distance $(\mbox{m})$ and $x y'$ is a distance $(\mbox{m})$. All three terms have the same dimension $(\mbox{m})$. This we call "equidimensional".