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Laplace Transforms, Inverse Laplace Transforms, and Differential Equations: Using MATLAB

This assignment provides a comprehensive analysis of Laplace transforms, inverse Laplace transforms, and differential equations, using MATLAB for verification.

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    Laplace Transforms, Inverse Laplace Transforms, and Differential Equations: A Comprehensive Analysis and Solution Verification Using MATLAB Exercises: 1. Find the Laplace transforms of the following functions b y hand and verify results using Matlab. General Laplace transform table F(t) f(t) 1 1 𝑠 𝑡 𝑛 𝑛 ! 𝑠 𝑛 + 1 𝑒 𝑎𝑡 / 𝑒 − 𝑎𝑡 1 𝑠 − 𝑎 / 1 𝑠 + 𝑎 sin 𝑎𝑥 𝑎 𝑠 2 + 𝑎 2 cos 𝑎𝑥 𝑠 𝑠 2 + 𝑎 2 y 𝑦 ̃ 𝑦 ̕ ̕ s 𝑦 ̕ − 𝑦 ( 0 ) y̋ 𝑠 2 𝑦 ̄ − 𝑠𝑦 ( 0 ) − 𝑦 ̕ ( 0 )

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    1. 𝑥 ( 𝑡 ) = 𝑡 Hand calculation ɭ[x (t)] = ∫ 𝑥 ( 𝑡 ) 𝑒 − 𝑠𝑡 𝑑𝑡 ∞ 0 = ∫ 𝑡 𝑒 − 𝑠𝑡 𝑑𝑡 ∞ 0 Using differentiation by parts Let u=t and dv= 𝑒 − 𝑠𝑡 du=1 and also v will be v= − 𝑒 − 𝑠𝑡 𝑠 x(t)= ∫ 𝑡 𝑒 − 𝑠𝑡 𝑑𝑡 ∞ 0 = − 𝑡𝑒 − 𝑠𝑡 𝑠 ǀ 0 ∞ + 1 𝑠 ∫ 𝑒 − 𝑠𝑡 𝑑𝑡 ∞ 0 = 1 𝑠 2 𝑒 − 𝑠𝑥 ǀ 0 ∞ =0 -- 1 𝑠 2 = 1 𝑠 2 Matlab code syms t % Declare a symbolic variable t F=laplace(t) %Calculate Laplace transform, store in F

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