# Thread: Newton's Law of Cooling Derivation

1. ## Newton's Law of Cooling Derivation

x(t) = Temperature of an object at time t.
(k < 0)
T = surrounding cooler temperature.

Given:
(dx/dt) = k(x(t) - T)

Derive using differential equations and show it equals:
x(t) = T + Ce^kt
Where C is a constant.

Attempt:
dx = k(x(t) - T) dt

Then integrate both sides?

2. ## Re: Newton's Law of Cooling Derivation

If you are trying to separate the variables then pull x(t) onto the LHS with dx.

3. ## Re: Newton's Law of Cooling Derivation

You cannot integrate x(t) with respect to t because you do not yet know what specific function of t x is.
As Chi Sigma says, you can "pull x(t) onto the LHS with dx" and integrate
$\displaystyle \int \frac{dx}{x- T}= \int dt$

Note that T is a constant.

4. ## Re: Newton's Law of Cooling Derivation

Originally Posted by HallsofIvy
As Chi Sigma says,
I wish I was that good with D.E.s! ;-)

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# derivation of ndwtons law of cooling

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