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2. (3 points) Find the Laplace transform of each of the following functions: (a) f(t) = (t² + 1)U (t − 3)

2. (3 points) Find the Laplace transform of each of the following functions: (a) f(t) = (t² + 1)U (t − 3)

Advanced Engineering Mathematics
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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Find the LaPlace transform of the function.

**Problem 2.** *(3 points)* Find the Laplace transform of each of the following functions: (a) \( f(t) = (t^2 + 1) \mathcal{U}(t - 3) \) In this problem, you are asked to calculate the Laplace transform of a given function. The function \( f(t) \) is composed of a polynomial part \((t^2 + 1)\) and a unit step function \(\mathcal{U}(t - 3)\), which shifts the response of the polynomial starting at \(t = 3\). The unit step function \(\mathcal{U}(t - 3)\) equals 0 for \(t < 3\) and 1 for \(t \geq 3\). Therefore, the expression \((t^2 + 1)\mathcal{U}(t - 3)\) describes a function that is zero before \(t = 3\) and follows the curve \(t^2 + 1\) after \(t = 3\). To solve for the Laplace transform, use properties of the Laplace transform, such as the shifting theorem, to handle the influence of the unit step function.
Transcribed Image Text:**Problem 2.** *(3 points)* Find the Laplace transform of each of the following functions: (a) \( f(t) = (t^2 + 1) \mathcal{U}(t - 3) \) In this problem, you are asked to calculate the Laplace transform of a given function. The function \( f(t) \) is composed of a polynomial part \((t^2 + 1)\) and a unit step function \(\mathcal{U}(t - 3)\), which shifts the response of the polynomial starting at \(t = 3\). The unit step function \(\mathcal{U}(t - 3)\) equals 0 for \(t < 3\) and 1 for \(t \geq 3\). Therefore, the expression \((t^2 + 1)\mathcal{U}(t - 3)\) describes a function that is zero before \(t = 3\) and follows the curve \(t^2 + 1\) after \(t = 3\). To solve for the Laplace transform, use properties of the Laplace transform, such as the shifting theorem, to handle the influence of the unit step function.
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