I have a question regarding the 1st step of this following example. Can someone explain me why the author set up ??? = x-2t and r = t? Please explain it in details. What is the general method of setting up such transformation of variables? I'm only confused in this part. The remaining part is fine.

Example 1.9 Find the general solution of Ut + 2ux - u = t. Let & = x-2t, T = t. In these characteristic coordinates the equation becomes UT - U = T. Multiplying by e- gives aT (Ue T ) = Te-T. Integrating, Ue-T = Te-TdT = -(1 + T)e-T + g(8), where g is an arbitrary function. Transforming back to at variables then gives the general solution u(x, t) = -(1+ t) +g(x - 2t)et. 0