How to solve this ODE tT' +2T = 0.
I'm solving for T here, and I think that T' is actually T'(t), but I'm not sure because the above is how my instructor emailed it to us, but then I don't know why the independent variable is multiplying T' though.
I used the characteristic and got that T= ce^((-2/t)*t)  which I'm very certain is not right.

This is a separable DE. Rewrite it as $\dfrac{T'}{T}=-\dfrac{2}{t}$, and integrate.
We get
$$\ln |T|=-2\ln|t|+C.$$
Take the exponential of both sides. We get $|T|=\dfrac{K}{t^2}$. If you know an "initial" value (the value of $T$ at some given $t_0$, you can find $K$.
Remark: The $T$, as you expected, is supposed to be a function of $t$, and the $T'$ refers to differentiation with respect to $t$. It would have been clearer if the equation had been written as $t\dfrac{dT}{dt}+2T=0$. 
