Verifying Equation is Dimensionless

[onie mti]

i am given this equation

View attachment 2323
where L is the rod of length when light signals are applied at one end (x=0)
D>0 is the diffusivity T>0 a time constant, A amplitude of the applied signal(carriers per second), g(t) the oscillatory behavior of the signal and α> 0 a material constant.

i need to change the equation above into one where all the symbols are dimensionless.
where
T is taken as the unit time
α^-1 the unit of length
AT the carrier density.

i am given that
the new dependent variables q=q(y,τ) is defined TAq(y, τ)= p(y/α,Tτ)
my work

from the above information, i defined new dimensionless independent variables
τ(tau) := t/ T(time)
y:= αx(position)

now how do i verify that the equation below is dimensionless and explain the symbols d and h(τ)
View attachment 2324

 

[gtoguy97]

L(x,t) = A g(t) h(τ(t)) + d(y,τ)To verify that the equation is dimensionless we need to check if all of the terms can be expressed as a combination of the new dimensionless independent variables y=αx and τ=t/T. The term A g(t) h(τ(t)) can be expressed as A(T/AT)g(Tτ)h(τ). This is a combination of the new dimensionless independent variables and therefore it is dimensionless. The term d(y,τ) can also be expressed as d(αy,Tτ) which is again a combination of the new dimensionless independent variables and therefore it is also dimensionless. Thus, we can conclude that the equation L(x,t) = A g(t) h(τ(t)) + d(y,τ) is indeed dimensionless. The symbols d and h(τ) refer to some arbitrary functions which are dimensionless.