By Mejlbro L.
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Additional info for Calculus 4c-2, Examples of Eigenvalue Problems
1) Prove that λ = 0 is not an eigenvalue. 2) Find all the eigenvalues and the corresponding eigenfunctions. 1) If λ = 0, then the equation is reduced to y = 0, the complete solution of which is y = c 1 + c2 x where y = c2 . Then by insertion into the boundary conditions, y(0) − y (0) = c1 − c2 = 0, hence c1 = c2 , and y(π) − y (π) = c1 + c2 π − c2 = 0, so c1 = −(π − 1)c2 . Since c1 = c2 = 0 is the only solution, we conclude that λ = 0 is not an eigenvalue. com 58 Examples of Eigenvalue Problems Eigenvalue problems 2) If λ = 0, then the characteristic polynomial R2 + 2λR + 2λ2 = (R + λ)2 + λ2 , has the simple roots R = −λ ± iλ.
We get from y(0) = 0 that c1 = 0, so we need only in the following to consider functions of the form (2) y(x) = c2 e−x sin(kx). We get from y(1) = 0 that 1 · sin k = 0. e We get proper solutions c2 = 0, when kn = nπ, n ∈ N. com 36 Examples of Eigenvalue Problems Eigenvalue problems • Eigenvalues and eigenfunctions. The eigenvalues are λn = 1 + kn2 = n2 π 2 + 1, n ∈ N. A corresponding eigenfunction is by (2) given by yn (x) = e−x sin(nπx), x ∈ [0, 1], Please click the advert and all eigenfunctions corresponding to λn are given by c · yn (x), where c = 0 is an arbitrary constant.
Now, ϕ(0) = 0, so sinh(k) − k cosh(k) < 0 for alle k > 0, and we only get the solution c2 = 0. Thus, no λ < 0 can be an eigenvalue. Alternatively we see that (1) is equivalent to tanh(k) = k, where a graphical analysis shows that k = 0 is the only solution. com 32 Examples of Eigenvalue Problems Eigenvalue problems 2) Let λ = 0, so the equation is reduced to d2 y = 0. dx2 • The complete solution follows by two integrations, y = c1 x + c 2 where y = c1 . • Insertion into the boundary conditions: y(0) = 0 = c2 , thus y = c1 x where y = c1 .
Calculus 4c-2, Examples of Eigenvalue Problems by Mejlbro L.