What definition do you have of a generalized function?
I mean the limiting formulation like these:
Dirac delta function - Wikipedia, the free encyclopedia
However I think a better way would be to use Laplace transforms to do this: have you used these before?
In laplace transforms you have the shift theorem which is referenced on the wiki site:
Laplace transform - Wikipedia, the free encyclopedia
Consider transforming both to Laplace space (frequency domain) and use the time-scaling and shift theorem to show that they are both equal Laplace transform functions.
Once you show that they are equal, then it follows that the functions are also equal.