Solve Heat ODE Modeling Problem: u(t) & x(t)

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The discussion centers on deriving a differential equation for the temperature of a heat plate, modeled as x(t), in response to an input power u(t). The user seeks help with formulating the equation, noting that the temperature change in the surrounding air can be neglected. The relevant heat balance leads to the equation x'(t) = - (g / C)x + (1 / C)u, where C is the heat capacity and g is the heat transfer coefficient. The user also inquires about better methods for typing equations, discovering LaTeX as a solution. The conversation concludes with the correct formulation of the differential equation for the heat plate problem.
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Im stuck on this question, can someone please help me?

u(t) = input power [W].
x(t) = temperature in plate [Celsius]
v = 0, temperature of surroundings [Celsius]
C = 400, heat capacity for plate [J/ Celsius]
g = 2, heat transfer plate / air [W / Celsius]

Question is something like this:
You're playing with the heat plate (Kitchen). The plate might be considered like a flat heat element, that radiates heat to the surroundings.
Find a differential equation for x(t).

Nb: the change of temperature in the room migtht be neglected due to
air circulation

Should be on the form like: x'(t) = ax + bu

I assume i need to use the ΔE = W + Q, but i kind of have no clue how to even start.
I checked if there was a solutions manual but, it was partly broken. The only experience i have with modelling something with heat is the heat equation. But that's a partial differential equation, and only one-dimentional, and i 'm not sure how to handle two functions in one equation?
 
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I know should try to come up with a possible solution, but I’ve been stuck with this all day, and I can't find any similar problems to start with either.
 
This is a 1D steady state heat conduction problem with heat generation within the body and a convective heat loss boundary condition at one of the surfaces. You need to perform a differential heat balance on a portion of the plate between x and x + Δx.
 
Ok, thanks.
 
Ok, I think I got it.

C dx = u dt - g(x-v)dt
x'(t) = - (g / C)x + (1 / C)u

Just an additional question: Is there some better way to type in equation, like Latex or similiar?
 
CyberneticsInside said:
Ok, I think I got it.

C dx = u dt - g(x-v)dt
x'(t) = - (g / C)x + (1 / C)u

Just an additional question: Is there some better way to type in equation, like Latex or similiar?
Do you see the words "LaTex / BBcode Guides" under the reply window?
 
Ah, found it !

So the answer would be \dot{x} = - \frac{g}{C}x + \frac{1}{C}u
 
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