# Newton's Law of Cooling Derivation

• Oct 18th 2011, 05:54 PM
ThatPinkSock
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?
• Oct 18th 2011, 06:07 PM
pickslides
Re: Newton's Law of Cooling Derivation
If you are trying to separate the variables then pull x(t) onto the LHS with dx.
• Oct 19th 2011, 01:17 PM
HallsofIvy
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
$\int \frac{dx}{x- T}= \int dt$

Note that T is a constant.
• Oct 19th 2011, 01:23 PM
pickslides
Re: Newton's Law of Cooling Derivation
Quote:

Originally Posted by HallsofIvy
As Chi Sigma says,

I wish I was that good with D.E.s! ;-)