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

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SUMMARY

The discussion focuses on solving the inhomogeneous heat equation for a cylindrical rod, represented by the equation ∂θ/∂t = D∇²θ + K. The objective is to derive the ordinary differential equation for θ(r), where R is the rod's radius and K signifies a constant heat production rate. The initial attempt using separation of variables was unsuccessful due to the presence of the constant K, indicating the need for alternative methods to address the inhomogeneity.

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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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Welcome to PF!

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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