Differential Equations Question (tricky)

A metal rod is 60 cm long and is heated at one end. The temperature at a point on the rod at distance x cm from the heated end is denoted by T. At a point halfway along the rod, T=290 and dT/dx = -6

(A) In a simple model for the temperature of the rod, it is assumed that dT/dx has the same value at all points on rod. For this model, express T in terms of X and hence determine the temperature difference between the ends of the rod.

(B) In a more refined model, the rate of change of T with respect to X is taken to be proportional to x. Set up a different equation for T, involving a constant of proportionality k. Solve the differential equation and hence show that, in this refined model, the temperature along the rod is predicted to vary from 380 degrees to 20 degrees


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I thought this was a PDE ...

A metal rod is 60 cm long and is heated at one end. The temperature at a point on the rod at distance x cm from the heated end is denoted by T. At a point halfway along the rod, T=290 and dT/dx = -6

(A) In a simple model for the temperature of the rod, it is assumed that dT/dx has the same value at all points on rod. For this model, express T in terms of X and hence determine the temperature difference between the ends of the rod.

It says dT/dx is the same all point and you know that dT/dx = -6 at one point, so you have that $\displaystyle \frac{dT}{dx} = -6$ which is a D.E you can solve. You'll need to use the fact that at x = 30, T=290 to find the arbitrary constant so you can compute $T(60) - T(0)$

Thanks got t=470-6x don't quite understand the end bit of your reply though.


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Assuming what you said is correct. The question then asks you to find the different of the temperature at the two ends of the rods. Temp at one end is temp at a distance x =60, this is given by T(60). Temp at other end is temp at a distance x=0, this ive given by T(0). Difference is T(60) -T(0)

Thanks! For part b what does it mean by rate of change being 'proportional' does that mean dT/dx = kX?


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Yes. Use the given information to find what the value of k is. (halfway along the rod, x =30 so -6 = k(30)

I solved dT/dx = -5x however I got T = -5x^2 + 5080 which is different to the answer as they used dT/dx = -kx I tried using that too however I still got something completely different.


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I'm not sure what you're saying? Remember that: $\int 5x = \frac{5x^2}{2}$

Oh I see, got confused by the dT and dX variables that I integrated it as -1/5 dt


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Does it work out now?

Ermmm no got the same thing as before 😩 answer is T = 380 - 1/10x^2


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k = -1/5 not -5.

?? -6(-5) = 30


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