Why Is My Solution to Newton's Law of Cooling Equation Incorrect?

[animboy]

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?

 

[Curious3141]

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:

$T(t) = T_m + e^{-kt}(T_0 - T_m)$

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.