I haven't done Laplace transforms in a while and I wanted to know if I did this right. I start out with the expression
$$\tau\frac{dT}{dt}+T(t)=T_{a}$$
I took the Laplace of this expression and got $$\tau[sT(s)-T(0)]+T(s)=\frac{T_{a}}{s}$$
which simplified to $$T(s)[s\tau+1]-\tau T(0)=\frac{T_{a}}{s}$$
Upon further simplification I wound up with
$$T(s)=\frac{T_{a}+\tau sT(0)}{s\tau+1}$$
would this be the right final answer? Again, I have not done Laplace in a while and I may have messed up algebraically somewhere so I just wanted some help on this. Thanks everyone