animboy said:
Homework Statement
dT/dt = -k(T - T_m)
T is the temperature of the body,
T_m is the temperature of the surroundings,
-k is some contant
and t is ofcourse time
Homework Equations
no idea
The Attempt at a Solution
I tried solving this using first order linear ODE integrating factor method:
so in standard form it can be written as-
T' + kT = T_m
let p(t) = k
then u(t) = exp(∫ k dt)
u(t) = exp(kt) we can forget about the constant.
so multiplying ODE throughout by u(x) :
exp(kt)*(T' + kT) = exp(kt)*(T_m)
integrate both sides
exp(kt)*T = [exp(kt)*(T_m)]/k
divide both sides by u(t)
T = (T_m)/k
my book says this is wrong, why?
A few issues:
Going from here: dT/dt = -k(T - T_m)
to here: T' + kT = T_m
you missed out a factor of k on the RHS.
Your equation should be T' + kT = kT_m.
Your Integrating Factor is OK (although this can be simply solved by separation of variables). But when you do the integration, you didn't put in bounds, which is a very important step when solving a physical problem.
Both sides are being integrated from t = 0 to t (the latter just represents a general time 't').
The temperature should be represented by T_0 initially, and T(t) at time t.
Evaluate both sides as definite integrals with the correct bounds and expressions for T, and you should get this result:
[itex]T(t) = T_m + e^{-kt}(T_0 - T_m)[/itex]
from which you can easily deduce the physical behaviour of the cooling body and see that it matches up to intuition, starting at T_0 and ending at infinite time at T_m.
As I mentioned, you don't need IF to solve this, you can just use separation of variables; it's less prone to error. Try it.