MHB Finding T with Newton's Cooling Law

[notagenuis]

Hi,

I just need help getting started with this problem. I am given the Newton's cooling law which is
T'= -k[T-Ta]
and I'm also given that Ta is a function also and it is Ta(t)=65-10cos(t)
I'm asking to solve for T when k=2 and T(0) = 75.

Do I just plug in 2 for k and plug Ta in the T'= function?

If so, I would get:
T'=-2T+130-20 cost (t)

Is this correct so far?

 

[Ackbach]

Hi,

I just need help getting started with this problem. I am given the Newton's cooling law which is
T'= -k[T-Ta]
and I'm also given that Ta is a function also and it is Ta(t)=65-10cos(t)
I'm asking to solve for T when k=2 and T(0) = 75.

Do I just plug in 2 for k and plug Ta in the T'= function?

If so, I would get:
T'=-2T+130-20 cost (t)

Is this correct so far?

Yes, it is correct so far. What kind of DE is this? How would you go about solving it?

 

[notagenuis]

Ok, i have attempted to solve it using an integration factor, right?

So
T'=-k[T-65+10cos(t)]
Im asked to assume that k=2 so i got
T'=-2T+130-20cos(t)
T'+2T=-20cos(t)+130
U(t)=e^(2t)
..
T=-4(sin(t)+2cos(t))+65+C
Initial condition T(0)=75 so when i plug it in i get
T=-4sin(t)-8cos(t)+18

Can you tell me if this looks good so far?

 

[Ackbach]

Ok, i have attempted to solve it using an integration factor, right?

So
T'=-k[T-65+10cos(t)]
Im asked to assume that k=2 so i got
T'=-2T+130-20cos(t)
T'+2T=-20cos(t)+130
U(t)=e^(2t)
..
T=-4(sin(t)+2cos(t))+65+C

The problem here is that when you multiply through by $e^{-2t}$ after integrating, the factor should hit the arbitrary constant. That is, you should have
$$T(t)=65-8\cos(t)-4\sin(t)+Ce^{-2t}.$$

Initial condition T(0)=75 so when i plug it in i get
T=-4sin(t)-8cos(t)+18

Can you tell me if this looks good so far?

Try propagating these changes.

 

[notagenuis]

Thank you very much. I followed your suggestions, this is what my answer looks like:

T=75e^(-2t) +65-8cost-4sint

I understand that the term 75e^(-2t) is a decaying term because it goes to 0 as t goes to infinity. However, how do you think I must describe the behavior of the function with an explicit formula? I understand that it's really the forcing term (65-the rest of the answer) is what's left...but how do I describe this in an explicit formula?

Thank you very much for your help walking me through this.

 

[Ackbach]

Thank you very much. I followed your suggestions, this is what my answer looks like:

T=75e^(-2t) +65-8cost-4sint

Hmm. With that equation for $T$, I get that
$$T(0)=75+65-8=132.$$
Are you sure you've computed $C$ correctly?

I understand that the term 75e^(-2t) is a decaying term because it goes to 0 as t goes to infinity.

Yep!

However, how do you think I must describe the behavior of the function with an explicit formula?

Not quite sure I understand what you're asking here. If you have the correct $T(t)$, then that IS the explicit formula! You can't get more explicit, in fact.

I understand that it's really the forcing term (65-the rest of the answer) is what's left...but how do I describe this in an explicit formula?

Oh, I suppose you could call the $Ce^{-2t}$ the transient behavior, or the forced response (forcing term or particular solution). Then you also have the natural response (homogeneous solution). It depends on if you're in an engineering environment or a mathematical one.

Thank you very much for your help walking me through this.

You're quite welcome!