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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

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    What is $\tau$? Is it a constant?2012-11-12
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    Is $T_a$ also just a constant?2012-11-12
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    Yes I am sorry. This is the equation for an object's response to a step change in temperature. So $T_{a}$ is a constant as the ambient temperature and $\tau$ is the time constant. They are both just constants2012-11-12

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