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-(2xy')'+ry= = Xy, y(1)=0, y(2)=y((2), = for x E [1,2]. (a) Is this a Sturm-Liouville problem? If so, explain why and decide whether it is regular or singular. If not, explain why not. (b) Suppose the boundary conditions were replaced by y(1) = 0, y(2) - y (1) = 0. Would this be a Sturm-Liouville problem? Explain your answer. Marks: 5 2. Consider the BVP Y"==x, y(0) = 0, y(1) = 0. (a) Find the associated Sturm-Liouville eigenvalue problem. (b) Find the eigenvalues and eigenfunctions. (c) Solve the BVP using an eigenfunction expansion. (d) Optional (not for marks): Solve the BVP directly and plot both solutions on the same axes. Hint: Pay careful attention to all signs. Marks: 12 3. Consider the integral equation So x. (a) What type of integral equation is this? State both the name and whether it is of the first or second kind. 1 of 2 (b) Suppose that T l. 0 (x + and T lo for l1 # À2. Find (211,u2). Make sure to clearly justify your answer. Marks: 4

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