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Notice that this eigenproblem is a … Obtain the eigenfunctions in x, Gn(x), that satisfy the PDE and boundary conditions (I) and (II) Step 2. For each equation we can write the related homogeneous or complementary equation: y′′+py′+qy=0. The method of separation of variables needs homogeneous boundary conditions. f ′′(x)=0 in this problem). For example, these equations can be written as ¶2 ¶t2 c2r2 u = 0, ¶ ¶t kr2 u = 0, r2u = 0. See also this post. It has a corresponding homogeneous equation a … Here, each λ k = kπ L 2 and φ k(x) = sin kπ L x is a eigenpair for the eigenproblem d2φ dx2 = −λφ for 0
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