Consider the following problem:

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c$($x$)\backslash$rho $($x$)$ut $=($K$0($x$)$ux$)$x $+$ Q$0,0<$ x $<$ L$,$

ux$(0)=0,$

ux$($L$)=0,$

where Q$06=0$ is a constant.

$($a$)$ Show that there does not exist any equilibrium temperature distribution. Briefly explain physically.

$($b$)$ Calculate the total thermal energy.