Did I use Laplace correctly? 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
 A: What you have done so far looks correct. However, I would strongly suggest that when you take the Laplace transform of the function $T(t)$, you use a new function symbol to indicate the Laplace transform, rather than just changing the variable from $t$ to $s$. The reason for this are many-fold; in final answer, for example, you have a $T(0)$; is that $T(t = 0)$ or $T(s = 0)$? It is not clear from the notation.
If you are trying to solve the differential equation for $T(t)$, once you have your final expression for the Laplace transform, then you have to use the inverse Laplace transform to obtain the solution, and taking the inverse Laplace transform is not always easy. If your homework asks you to take the Laplace transform, then I suppose that is what you'll need to do, but if you're just trying to solve the differential equation, then the Laplace transform is not the best way, because your differential equation is just an inhomogeneous linear ODE with constant coefficients, so it can be solved by more classical methods, like using the method of undetermined coefficients, for example.
