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?

Re: Newton's Law of Cooling Derivation

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

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.

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! ;-)