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

u(1)=u(e)=0 .

\end{array}\right. (ii) Find the solution to the boundary value problem (2). Hint: Using the orthonormal sequence {phi}n>1 of eigen functions of (1) from Question 3(b), look for a solution of the form u(x) = Σ₂₂(1) and determine the coefficients , using generalized Fourier series. (b) Let ƒ € C¹([1, ∞0)) satisfy ƒ(1) = 1 and ƒ(e) = 0. Consider the following n on-homogeneous initial boundary value problem ) Find the constants a and b such that u(x,t) = w(x, t) + az + b solvesthe corresponding homogeneous equation u_{\mathrm{t}}+u-x\left(x u_{x}\right)_{x}=0 \quad \text { for } 1

) Justify why problem (2) has at most oneC²-solution on [1, e].

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