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\alpha^2T_{xx} = T_t + \beta(T - T_0)

\]

where \(\beta\) is a constant and \(T_0\) is the temperature of the surrounding medium. The initial temperature distribution is \(T(x, 0) = f(x)\) and the ends \(x = 0\) and \(x = \ell\) are maintained at \(T_1\) and \(T_2\) when \(t > 0\).

Show that the substitution \(T(x, t) = T_0 + U(x, t)e^{-\beta t}\) reduces the problem to the following one:

\[

\alpha^2U_{xx} = U_t

\]

with new initial conditions and boundary conditions for \(U\).

With that substitution, I obtain:

\begin{align}

\alpha^2U_{xx} &= -\beta(U_t - T_0U)\\

\alpha_1^2U_{xx} &= U_t - T_0U

\end{align}

What is going wrong?