4s²-5s+6 (s+1)(s²+4) 2- Inverse Laplace transform for a) f(t) = 3et+ cos2t - 3 sin 2t b) f(t) = et + cos2t + 3 sin 2t c) f(t) = 3e-t- cos 2t + 3 sin 2t d) f(t) = et + cos2t - 3 sint is:
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- In the integration of (cos x)^4 (sin x)^3 dx, what is the appropriate transformation? * A.) All options are incorrect B.) Transform (cos x)^2 to 1/2(1 + cos 2x) C.) Transform (cos x)^2 to 1 - (sin x)^2 D.) Transform (sin x)^2 to 1 - (cos x)^2Suppose U solves the heat equation on the real lineUt = 4Uxx, x ∈ Rwith initial valueU(x, 0) = (4, x ≤ 02, x > 0.(i) Use the Fourier-Poisson formula to give an explicit expression for the solutionU.(ii) Describe the qualitative behaviour of U in this case as t → ∞ and plot outthe solution at several instants of time to explain your answer. What is the limitof U as t → ∞?solve for x and y of the given ODE using laplace transform.please provide complete/detailed solution.