Find bases for the kernels of the following linear trans- formations from Co(-∞, ∞) to C∞ (-∞, ∞). 14. Dª +2D² +1
Find bases for the kernels of the following linear trans- formations from Co(-∞, ∞) to C∞ (-∞, ∞). 14. Dª +2D² +1
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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![**Problem Statement:**
Find bases for the kernels of the following linear transformations from \( C^\infty(-\infty, \infty) \) to \( C^\infty(-\infty, \infty) \).
**Task 14:**
\[ D^4 + 2D^2 + 1 \]
**Explanation:**
The task involves finding the kernel of a linear differential operator applied to functions within the space of infinitely differentiable functions over the entire real line. The operator given, \( D^4 + 2D^2 + 1 \), is a composition of derivatives where \( D \) denotes the differentiation operator.
**Objective:**
Determine the set of functions \( f(x) \) such that the operator applied to \( f(x) \) equals zero:
\[ (D^4 + 2D^2 + 1)f(x) = 0 \]
This involves solving a fourth-order differential equation and finding a basis for the solution space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbfa44710-6742-4cf8-9a38-b63fd4325c9b%2Fe9084d44-ae07-4db8-9c02-b24d20f49fc4%2F9jzb0eg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find bases for the kernels of the following linear transformations from \( C^\infty(-\infty, \infty) \) to \( C^\infty(-\infty, \infty) \).
**Task 14:**
\[ D^4 + 2D^2 + 1 \]
**Explanation:**
The task involves finding the kernel of a linear differential operator applied to functions within the space of infinitely differentiable functions over the entire real line. The operator given, \( D^4 + 2D^2 + 1 \), is a composition of derivatives where \( D \) denotes the differentiation operator.
**Objective:**
Determine the set of functions \( f(x) \) such that the operator applied to \( f(x) \) equals zero:
\[ (D^4 + 2D^2 + 1)f(x) = 0 \]
This involves solving a fourth-order differential equation and finding a basis for the solution space.
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