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Problem 8. Suppose p E P(R). Prove that there exists a polynomial q E P(R) such that 5q" +3q' p. Here, q' denotes the derivative of q. = Hint: Observe that Tq := 5q" + 3q' is a linear map. What is its kernel?

Problem 8. Suppose p E P(R). Prove that there exists a polynomial q E P(R) such that 5q" +3q' p. Here, q' denotes the derivative of q. = Hint: Observe that Tq := 5q" + 3q' is a linear map. What is its kernel?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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**Problem 8.** Suppose \( p \in \mathcal{P}(\mathbb{R}) \). Prove that there exists a polynomial \( q \in \mathcal{P}(\mathbb{R}) \) such that \( 5q'' + 3q' = p \). Here, \( q' \) denotes the derivative of \( q \). **Hint:** Observe that \( Tq := 5q'' + 3q' \) is a linear map. What is its kernel?
Transcribed Image Text:**Problem 8.** Suppose \( p \in \mathcal{P}(\mathbb{R}) \). Prove that there exists a polynomial \( q \in \mathcal{P}(\mathbb{R}) \) such that \( 5q'' + 3q' = p \). Here, \( q' \) denotes the derivative of \( q \). **Hint:** Observe that \( Tq := 5q'' + 3q' \) is a linear map. What is its kernel?
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