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2. Determine the Laplace transform of the given function a) f(t) = e³¹t4 b) f(t

2. Determine the Laplace transform of the given function a) f(t) = e³¹t4 b) f(t

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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**Problem Statement:** Determine the Laplace transform of the given function. **Functions:** a) \( f(t) = e^{3t} t^4 \) b) \( f(t) = 2e^{3t} - 3te^{2t} \) c) \( f(t) = e^{5t} \sin(3t) \) d) \( f(t) = e^{-4t} \cosh(2t) \) e) \( f(t) = e^{5t} \cos^2(5t) \) f) \( f(t) = e^t \cos t + e^t \sin^2(3t) \) g) \( f(t) = (t - 3)^2 U_3(t) \) h) \( f(t) = e^{5(t-2)} U_2(t) \) i) \( f(t) = t^2 U_3(t) \) j) \( f(t) = e^{5(t-4)} (t - 1)^2 U_4(t) \) k) \( f(t) = t^3 \cos(3t) \) l) \( f(t) = te^{3t} \sin(2t) \) **Explanation:** These functions are generally given in terms of \(t\) and include exponential functions, trigonometric functions, and step functions \(U(t)\). The Laplace transform is a powerful integral transform used to convert differential equations to algebraic equations, making them easier to solve.
Transcribed Image Text:**Problem Statement:** Determine the Laplace transform of the given function. **Functions:** a) \( f(t) = e^{3t} t^4 \) b) \( f(t) = 2e^{3t} - 3te^{2t} \) c) \( f(t) = e^{5t} \sin(3t) \) d) \( f(t) = e^{-4t} \cosh(2t) \) e) \( f(t) = e^{5t} \cos^2(5t) \) f) \( f(t) = e^t \cos t + e^t \sin^2(3t) \) g) \( f(t) = (t - 3)^2 U_3(t) \) h) \( f(t) = e^{5(t-2)} U_2(t) \) i) \( f(t) = t^2 U_3(t) \) j) \( f(t) = e^{5(t-4)} (t - 1)^2 U_4(t) \) k) \( f(t) = t^3 \cos(3t) \) l) \( f(t) = te^{3t} \sin(2t) \) **Explanation:** These functions are generally given in terms of \(t\) and include exponential functions, trigonometric functions, and step functions \(U(t)\). The Laplace transform is a powerful integral transform used to convert differential equations to algebraic equations, making them easier to solve.
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