Question

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

-\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 } 10 subject to u(1, t)=u(e, t)=0 \quad \text { for } t>0 . Solve the initial boundary value problem corresponding to u(x, t) using the method of separation of variables and Question 3. (Write the Fourier coefficients in integral form without further calculation.) Hence, find a solution w of (3).

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

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