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Consider the boundary value problem \left\{\begin{array}{l}

-\left(x u_{x}\right)_{x}+\frac{2 u}{x}=\frac{5 \ln x}{x} \text { for } 1

u^{\prime}(1)=u^{\prime}\left(e^{x}\right)=0 .

\end{array}\right. (i) Justify why problem (2) has at most one C²-solution on [1, eª].

) Consider the following non-homogeneous initial boundary value problem

We denote by U₁(r) the solution of (2) found in (ii). Find a func-tion U₂(t) (depending only on t) satisfying U₂(0) = 0) and such thatu(x, t) = U₁(x) + U₂(t) + w(x, t) solves the homogeneous equation u_{t}+2 u-x\left(x u_{x}\right)_{x}=0 \text { for } 10 ii) Solve the corresponding initial boundary value problem for (r, t) us-ing the method of separation of variables and Question 3. Hence, find a solution w of (3).

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