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physstudent.4

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## Homework Statement

du/dt=(d^2 u)/dx^2+1

u(x,0)=f(x)

du/dx (0,t)=1

du/dx (L,t)=B

du/dt=0

Determine an equilibrium temperature distribution. For what value of B is there a solution?

## Homework Equations

Not really sure what to put here.

## The Attempt at a Solution

I started by trying to separate variables, with u(x,t)=phi(x)*g(t), and got to

g'(t)/g(t)=phi''(x)/phi(x)+1/(phi(x)g(t))=0.

So g(t) is constant based on the above, but then I get a little lost while trying to solve for phi. I tried letting g(t)=lamda (abbreviated lm from now on), and got

phi''(t)+1/lm=0, which yields a quadratic solution of [(-lm*x^2)/2+x/lm+C/lm] after using the condition du/dx (0,t)=1. Then, since this is phi(x),

u(x,t)= -lm^2*x^2/2+ x + lm*C.

Using the other condition du/dx (L,t)=B, and assuming some quick mental algebra was correct,

B=-lm^2*L+1.

First off, is all of the above a correct approach as far as you can tell? And secondly, do I need to find u(x,t) or a specific value of lm?

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