Find the Laplace transform of the following functions (a) sin(5t) * e−2t (here star is convolution) 0 t< 4 (b) g(t) = 1 t 4. > 5 t<4 (c) g(t): = (d) g(t) = 0 t≥ 4. (5 t<4 -1 t> 4. [Hint: this is a linear combination of two step functions]

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A: We have to solve given problem:

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Q: 7. Using Lyapunov functions, show that the origin is unstable for the ODE i = -x3 ÿ = -y3

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A: Laplace transform of the function

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Q: 1.a Find the Laplace transform of the function F(t) = e-2t (cosh 3t + sinh 5t)

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A: Sol:-

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