How to Solve the Inhomogeneous Heat Equation for a Cylindrical Rod?

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The discussion centers on solving the inhomogeneous heat equation for a cylindrical rod, specifically the equation ∂θ/∂t = D∇²θ + K, where K represents a constant heat production rate. The challenge is to derive the ordinary differential equation for θ(r) given the rod's radius R. Initial attempts using separation of variables were unsuccessful due to the presence of the constant K. Participants suggest reviewing resources on the homogeneous heat equation for guidance. The conversation emphasizes the need for a tailored approach to account for the inhomogeneity in the equation.
mumaga
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inhomogeneuos heat equation!

Homework Statement


∂θ/∂t= D∇2θ + K, the mensioned equation is the heat equation for a cylindrical rod , and the requaired is to find the ordinary differential equation for θ(r) .where the radius of the rod is R , and K is constant ( correspond to a constant rate pf heat production)



Homework Equations





The Attempt at a Solution


i use separation of variables to obtain the required for a homogeneous heat equation , but with the constant the method didn't work out.
thanks for your time.
 
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mumaga said:
∂θ/∂t= D∇2θ + K, the mensioned equation is the heat equation for a cylindrical rod , and the requaired is to find the ordinary differential equation for θ(r) .where the radius of the rod is R , and K is constant ( correspond to a constant rate pf heat production)

Hi mumaga! Welcome to PF! :smile:

(I assume you mean ∂θ/∂t= C∇2θ + K, where C and K are constants, and θ depends only on t and r.)

See http://en.wikipedia.org/wiki/Heat_equation#Homogeneous_heat_equation :smile:
 

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