, Let C" (R) be the vector space of "smooth" functions, i.e., real-valued functions f(x) in the variable z that have infinitely many derivatives at all points x E R. Let D : C" (R) →→ C®(R) and D2 : C¤(R) → C®(R) be the linear transformations defined by the first derivative D(f(x)) = f'(x) and the second derivative D²(f(x)) = f"(x). a. Determine whether the smooth function g(x) = 7e-lz is an eigenvector of D. If so, give the associated eigenvalue. If not, enter NONE. Eigenvalue = b. Determine whether the smooth function h(x) = sin(9x) is an eigenvector of D2. If so, give the associated eigenvalue. If not, enter NONE. Eigenvalue =

, Let C" (R) be the vector space of "smooth" functions, i.e., real-valued functions f(x) in the variable z that have infinitely many derivatives at all points x E R. Let D : C" (R) →→ C®(R) and D2 : C¤(R) → C®(R) be the linear transformations defined by the first derivative D(f(x)) = f'(x) and the second derivative D²(f(x)) = f"(x). a. Determine whether the smooth function g(x) = 7e-lz is an eigenvector of D. If so, give the associated eigenvalue. If not, enter NONE. Eigenvalue = b. Determine whether the smooth function h(x) = sin(9x) is an eigenvector of D2. If so, give the associated eigenvalue. If not, enter NONE. Eigenvalue =

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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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

, Let C" (R) be the vector space of "smooth" functions, i.e., real-valued functions f(x) in the variable z that have infinitely many
derivatives at all points x E R.
Let D: C* (IR) → C¤(R) and D² : C∞ (R) → C°(R) be the linear transformations defined by the first derivative D(f(x)) = f'(x) and
the second derivative D²(f(x)) = f"(x).
a. Determine whether the smooth function g(x) = 7e1z is an eigenvector of D. If so, give the associated eigenvalue. If not, enter NONE.
Eigenvalue =
b. Determine whether the smooth function h(x) = sin(9x) is an eigenvector of D2. If so, give the associated eigenvalue. If not, enter
%3D
NONE.
Eigenvalue =

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