Newton's Law of Temperature Change Differential

[Manni]

I'm a bit confused as to what I do next with this problem.

Consider the initial value problem \frac{dy}{dx} = a(y-b) where y(0) = y₀

With a > 0, b = 0, it represents exponential growth, while a < 0, b = 0 gives exponential decay. With y = T, a = -k, k = T_A and y₀=T₀ it gives Newton's Law of Temperature Change,

\frac{dy}{dx} = -k(T - T_A)

Solve the given model and give a qualitative sketch for each of the cases 1) a > 0 and 2) a < 0. You may assume b ≥ 0. Include y₀= b as one typical solution for each.

So what I did was I applied the integral to both sides of \frac{dy}{dx} = -k(T - T_A) yielding,

1 = -k ∫ (T - T_A)

But I don't know how to integrate T_A

 

[HallsofIvy]

You can't integrate both sides because you have no differential on the right. Part of your problem is that you say y= T but then leave y in the equation.
Write the equation as \frac{dT}{dx}= -k(T- T_A), then as
\frac{dT}{T- T_A}= -kdx
and integrate. T_A is a constant. To integrate on the left, let u= T- T_A so that du= dT.

 

[Ray Vickson]

I'm a bit confused as to what I do next with this problem.

Consider the initial value problem \frac{dy}{dx} = a(y-b) where y(0) = y₀

With a > 0, b = 0, it represents exponential growth, while a < 0, b = 0 gives exponential decay. With y = T, a = -k, k = T_A and y₀=T₀ it gives Newton's Law of Temperature Change,

\frac{dy}{dx} = -k(T - T_A)

Solve the given model and give a qualitative sketch for each of the cases 1) a > 0 and 2) a < 0. You may assume ≥ 0. Include y₀= b as one typical solution for each.







So what I did was I applied the integral to both sides of \frac{dy}{dx} = -k(T - T_A) yielding,

1 = -k ∫ (T - T_A)

But I don't know how to integrate T_A

I think you are mis-reading the problem. It says to solve the given model, and to look at the cases a > 0 and a < 0, etc. To me, that says solve dy/dx = a*(y-b) [= the given model], so there is no "T" involved.

RGV

 

[Manni]

Oh, so we would integrate the constant b as a normal constant correct? I.e. b*x?

 

[HallsofIvy]

Not when it is in the denominator! Use the substitution u= T- T_A as I suggested.

 

[Manni]

Oh I understand, thank you!

 

[Manni]

Wait, it's dT/dt not dT/dx?

 

[Manni]

The only thing I'm not sure of is if it the question is asking for solutions. If it is can I put the dy/dx equation in the standard differential form,

dy/dx + P(x) = Q(x)

Then, I can determine the integrating factor and solve that way. I was initially thinking of doing that, would it work?