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Math Help - Differential equation question

  1. #1
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    Differential equation question

    Here's the instructions:

    In Exercises 37-39, use Newton's Law of Cooling, which states that the rate of change in the temperature T of an object is proportional to the difference between the temperature T of the object and the temperature T0 of the surrounding environment. This is described by the differential equation \frac{dT}{dt}=k(T-T_0)

    In the solutions manual they alter the DE to T=Ce^{kt}+T_0

    How do they change it like that? The section it's in is separation of variables so I've been getting all the x's on one side, all the y's on one side, etc. but I don't quite see how they do that at all in this instance...
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  2. #2
    Senior Member Peritus's Avatar
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    <br />
\frac{1}<br />
{k}\frac{{dT}}<br />
{{T - T_0 }} = dt<br />

    now integrate both sides:

    <br />
\begin{gathered}<br />
  \frac{1}<br />
{k}\ln \left( {T - T_0 } \right) = t + \beta  \hfill \\<br />
   \hfill \\<br />
   \leftrightarrow T - T_0  = e^{kt + k\beta }  \hfill \\<br />
   \hfill \\<br />
   \leftrightarrow T(t) = \underbrace {e^{k\beta } }_Ce^{kt}  + T_0  = Ce^{kt}  + T_0  \hfill \\ <br />
\end{gathered} <br />
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  3. #3
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    Quote Originally Posted by Peritus View Post
    <br />
\frac{1}<br />
{k}\frac{{dT}}<br />
{{T - T_0 }} = dt<br />

    now integrate both sides:

    <br />
\begin{gathered}<br />
  \frac{1}<br />
{k}\ln \left( {T - T_0 } \right) = t + \beta  \hfill \\<br />
   \hfill \\<br />
   \leftrightarrow T - T_0  = e^{kt + k\beta }  \hfill \\<br />
   \hfill \\<br />
   \leftrightarrow T(t) = \underbrace {e^{k\beta } }_Ce^{kt}  + T_0  = Ce^{kt}  + T_0  \hfill \\ <br />
\end{gathered} <br />
    You are a lifesaver, thank you so much!
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  4. #4
    GAMMA Mathematics
    colby2152's Avatar
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    Quote Originally Posted by emttim84 View Post
    You are a lifesaver, thank you so much!
    Tim, I believe you are much more of a lifesaver i.e. your signature being true!
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