Consider the set of all smooth functions on R as C° (R) := {ƒ : R → R | f(n) exists and it is continuous for all n € N}. A subset AC C(R) is finite dimensional if there exists a finite subset {f1, f2,... fa} of C(R) such that span{f1, f2,..., fa} = A; otherwise, A is infinite dimensional. Given a function f = C(R), define the set Df as Df := = {f(n) |ne NU {0}}. Problem 2 (a) For a given f = C(R), verify that Df C C (R). (b) For f = sin x + cos x, verify that Df is finite dimensional. Then, find a finite subset B C C (R) of linearly independent functions such that span B = Df.
Consider the set of all smooth functions on R as C° (R) := {ƒ : R → R | f(n) exists and it is continuous for all n € N}. A subset AC C(R) is finite dimensional if there exists a finite subset {f1, f2,... fa} of C(R) such that span{f1, f2,..., fa} = A; otherwise, A is infinite dimensional. Given a function f = C(R), define the set Df as Df := = {f(n) |ne NU {0}}. Problem 2 (a) For a given f = C(R), verify that Df C C (R). (b) For f = sin x + cos x, verify that Df is finite dimensional. Then, find a finite subset B C C (R) of linearly independent functions such that span B = Df.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Consider the set of all smooth functions on R as
C(R) := {ƒ : R → R | ƒ(n) exists and it is continuous for all n € N}.
A subset AC C(R) is finite dimensional if there exists a finite subset {f1, f2, ... fa} of C∞(R) such
that span{f1, f2,..., fa} = A; otherwise, A is infinite dimensional.
Given a function ƒ € C∞(R), define the set Df as
Problem 2
Dƒ := {f(n) | n = NU{0}}.
Df
(a) For a given ƒ € C°(R), verify that Dƒ C C≈(R).
(b) For f = sin x + cos x, verify that Dƒ is finite dimensional. Then, find a finite subset B C C∞(R)
of linearly independent functions such that span B = Dƒ.
(c) Consider the linear non-homogeneous equation
y" + y = e sinx.
(1)
Show that Df, where f = e sinx, is finite-dimensional; moreover, find a subset with two elements
B = {f1, f2} C C∞(R) of linearly independent functions such that span BD. CB
(d) Find a particular solution yp for equation (1) using the method of undetermined coefficients: Let
yp E span B i.e. there exists C₁, C2 ER such that yp = c₁f1 + c2f2; put yp into the equation (1) and
find the constants c₁ and c₂.
(Note that the method of undetermined coefficients is not suitable if f is not a sum of the products of
sin x, cos x, the exponential function, and polynomials of x some of these functions might be missing
in the products.)
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