MATH 467 Partial Differential Equations Exercises - Millersville ...

34. For the boundary value problem below, find all the values of L for which there exists
a solution.
y ′′ +y = 0 for 0 ≤ x ≤ L
y(0) = 0
y(L) = 1
35. For the boundary value problem below, show that there are infinitely many positive
eigenvalues {λn} ∞ n=1 where
36. Show that if a /∈ Z that
for −π < x < π.
πcos(ax)
2asin(aπ)
37. For 0 < x < 2π show that
e x = e2π −1
π
38. Use the result above to show that
lim
n→∞ λn = 1
4 (2n−1)2 π 2 .
y ′′ +λy = 0 for 0 ≤ x ≤ 1
y(0) = 0
y(1) = y ′ (1)
1 cosx
= +
2a2 12 cos(2x)
−
−a2 22 cos(3x)
+
−a2 32 −···
−a2
1
2 +
π cosh(π −x)
·
2 sinhπ
∞
n=1
cos(nx)−nsin(nx)
n2
.
+1
= 1
2 +
∞
n=1
cos(nx)
n 2 +1 .
39. Use the result above to find the sum of the infinite series
∞ 1
n2 +1 .
n=1
40. Use the result above to find the sum of the infinite series
∞ 1
(n2 +1) 2.
n=1