Find the eigenvalues and eigenfunctions for −y = λy on bounded interval

| August 31, 2017

Question
HW 3
Due ThursdayNov 5th

1. Find the eigenvalues and eigenfunctions for −y = λy on bounded interval −l < x < l assuming the
periodic boundary conditions y(−l) = y(l) and y (−l) = y (l). Follow the method used in class and be
sure to consider the case λ < 0, λ = 0 and λ > 0.
2. Solve the following 4-th order eigenvalue problem: y (4) = λy on the interval 0 < x < l assuming λ > 0
with the boundary conditions y(0) = y (0) = y(l) = y (l) = 0.
3. (a) Let u(x) and v(x) be two twice continuously differentiable functions. “Prove” the following
formula known as Green’s First Identity:
l

l

v (x)u(x) dx = [u(l)v (l) − u(0)v (0)] −
0

v (x)u (x) dx
0

(Hint: Integrate the left side of the equation by parts)
(b) Now consider the eigenvalue problem analyzed in class: −y = λy. Show directly, using Green’s
formula above, that under the boundary condition y(0) = y(l) = 0, the eigenvalues are non
negative. (Hint: Multiply the ODE by y and integrate and then apply Green’s formula)
(c) Show that the same conclusion holds if instead we assume the boundary condition y (0) = y (l) = 0
or the boundary condition y (0) = y(l) = 0.
(d) “Prove” the following formula known as Green’s Second Identity:
l

l

v (x)u(x) dx −
0

v(x)u (x) dx = [u(l)v (l) − u(0)v (0)] − [u (l)v(l) − u (0)v(0)]
0

(e) Show that, under any of the boundary conditions in parts (b) and (c), eigenfunctions corresponding to distinct eigenvalues are orthogonal. In other words, if λ1 and λ2 are eigenvalues with
l

corresponding eigenfunctions y1 (x) and y2 (x), show that if λ1 = λ2 then

y1 (x)y2 (x) dx = 0.
0

(f) In your own words and in no more than 5 sentences, summarize what you have shown and relate
it to what was done in class.
4. Let f (x) = π 2 − x2 on the interval (−π, π).
(a) Sketch the 2π-periodic extension of f
(b) Compute the Fourier Series of the extended function.
5. Find the Fourier series of:

−π

2x
f (x) =
π

0

x ∈ (π, − π )
2
x ∈ [− π , π ]
2 2
x ∈ ( π , π)
2
x = ±π

Be sure to include a sketch of the 2π-periodic extension of f .

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