find the partial fraction the fraction expression for 1 s2

find the partial fraction the fraction expression for 1 + s^2 - s^3 / (s^3-s)(s^2-) and on this basis.. Find the inverse Laplace L^-1 {P}

Solution

If the denominator could be factored nicely, we would use partial fractions. This denominator does not factor nicely (unless we use complex numbers). When that happens, try “completing the square” to rewrite the denominator in terms of “ s a ” for some constant a . Here, s 2 8s + 25 = s 2 2 · 4s + 4 2 4 2 + 25 = s 2 2 · 4s + 4 2 | {z } (s4) 2 4 2 + 25 = (s 4) 2 + 9 . Hence, L 1 1 s 2 8s + 25 t = L 1 1 (s 4) 2 + 9 t = L 1 [F(s 4)]| t = e 4t f (t) . (26.3) Again, we need to find f (t) from a shifted version of its transform. Here, F(s 4) = 1 (s 4) 2 + 9 . Additional Exercises 537 Letting X = s 4 in this equation, we have F(X) = 1 X2 + 9 , which means the formula for F(s) is F(s) = 1 s 2 + 9 . Thus, f (t) = L 1 [F(s)]| t = L 1 h 1 s 2 + 9 i t = L 1 h 1 3 · 3 s 2 + 9 i t = 1 3 L 1 h 3 s 2 + 3 2 i t = 1 3 sin(3t) . Plugging this back into equation (26.3), we get L 1 h 1 s 2 8s + 25 i t = · · · = e 4t f (t) = e 4t 1 3 sin(3t) = 1 3 e 4t sin(3t)