MATH 519 Homework
Fall 2013
16. Let uj (t) = tλj where λ1 , . . . λn are arbitrary unequal real numbers. Show that
{u1 . . . un } are linearly independent functions on any interval (a, b) ⊂ R. (Suggestion: If
P
n
λj
= 0, divide by tλ1 and differentiate.)
j=1 αj t
17. A side condition for a differential equation is homogeneous if whenever two functions
satisfy the side condition then so does any linear combination of the two functions. For
example the Dirichlet
type boundary condition u = 0 for x ∈ ∂Ω is homogeneous.
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Let Lu = |α|≤m aα (x)Dα u denote any linear differential operator. Show that the
set of functions satisfying Lu = 0 and any homogeneous side conditions is a vector space.
18. Consider the differential equation u00 + u = 0 on the interval (0, π). What is the
dimension of the vector space of solutions which satisfy the homogeneous boundary
conditions a) u(0) = u(π), and b) u(0) = u(π) = 0. Repeat the question if the interval
(0, π) is replaced by (0, 1).
19. If (X, dX ), (Y, dY ) are metric spaces, show that the Cartesian product
Z = X × Y = {(x, y) : x ∈ X, y ∈ Y }
is a metric space with distance function
d((x1 , y1 ), (x2 , y2 )) = dX (x1 , x2 ) + dY (y1 , y2 )
20. Show by examples that neither of the following conditions concerning a sequence of
nonnegative functions {fn }∞
n=1 implies the other one.
(i) limn→∞ fRn (x) = 0 for all x ∈ [0, 1]
1
(ii) limn→∞ 0 fn (x) dx = 0