1. Calculate the inverse Laplace transform c-1J(-2s – 4)e¬³ s2 + 6s + 25 4e¬2s (s + 1)3 2. Solve the initial value problem by the method of Laplace transforms: y" + 3y' + 2y = 6u(t – In 2), y(0) = 1,...

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Extracted text: 1. Calculate the inverse Laplace transform c-1J(-2s – 4)e¬³ s2 + 6s + 25 4e¬2s (s + 1)3 2. Solve the initial value problem by the method of Laplace transforms: y" + 3y' + 2y = 6u(t – In 2), y(0) = 1, y'(0) = –3 3. Solve the initial value problem by the method of Laplace transforms: y" + 225y = 308(t – 1), y(0) = 20, y'(0) = 0 4. Consider the following boundary value problem for y(x) on [–2, 2]: (*) y" + 6y' + dy = 0, y(-2) = 0, y(2) = 0. (a) Is y(x) eigenvalue. = e-3* cos () an eigenfunction of (*)? If it is, compute the corresponding (b) Is A= 10 an eigenvalue of (*)? If it is, calculate the corresponding eigenfunctions. (c) Determine all eigenvalues A satisfying A > 15 and calculate the corresponding eigenfunc- tions.