5.2.2. Define K : C [0, l] C[0, l] as d u Ku ==a + bu,...

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5.2.2. Define K : C [0, l] C[0, l] as d u Ku ==a + bu, dx2 where a and b are positive constants. We know (from Exercise 5.1.8) that K has only real, positive eigenvalues. Find all of the eigenvalues and eigenvectors of K. Solve the following BVPs using the method of Fourier series, shifting the data if necessary. If possible, produce a graph of the computed solution by plotting a partial Fourier series with enough terms to give a qualitatively correct graph. (The number of terms can be determined by trial and error; if the plot no longer changes, qualitatively, when more terms are included, it can be assumed that the partial series contains enough terms.) 5.2.2. Define K : C [0, l] C[0, l] as d u Ku ==a + bu, dx2 where a and b are positive constants. We know (from Exercise 5.1.8) that K has only real, positive eigenvalues. Find all of the eigenvalues and eigenvectors of K. Solve the following BVPs using the method of Fourier series, shifting the data if necessary. If possible, produce a graph of the computed solution by plotting a partial Fourier series with enough terms to give a qualitatively correct graph. (The number of terms can be determined by trial and error; if the plot no longer changes, qualitatively, when more terms are included, it can be assumed that the partial series contains enough terms.)