Solving recurrence using characteristic equation

I am recently learning the how to use the various methods to solve recurrences. So far I have acquainted myself with the Master's Theorem and Substitution method. One method I just can't seem to understand is the following question which needs to be solved by characteristic equation:

$$T(n) = 2T(n/3) + 1,T(1) = 1$$

I watched certain tutorials and readings and I imagine I needed to derive some sort of degree and simultaneous equation out of this?

How do I do it in this case?

Sorry for being amateur at this.

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• Are you sure it needs to be solved by characteristic equation only? – Beyond Infinity 2 days ago

Write $$n=3^m$$ so $$T(3^m)=2T(3^{m-1})+1$$.
Then let $$T(3^m)=u(m)$$ so $$u(m)=2u(m-1)+1$$.