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ENGR5001 / Week 3
ENGR201 Laplace deck example, partial fractions with a quadratic factorTutorialOld spec ENGR2012:112 min

ENGR201 (old spec) Laplace transforms deck, partial fractions case 3 (pages 38-39)

Find L−1[X(s)]\mathcal{L}^{-1}[X(s)] for X(s)=1s(s2+4s+13)X(s) = \frac{1}{s(s^2 + 4s + 13)} and evaluate the result at t=0.5t = 0.5. (This is the unit step response of 1/(s2+4s+13)1/(s^2 + 4s + 13).)

Formulas you may need
  • Quadratic factor with no real roots: term Bs+Cs2+ps+q\dfrac{Bs + C}{s^2 + ps + q}; complete the square (learn this)
  • L−1[s−a(s−a)2+b2]=eatcos⁡bt\mathcal{L}^{-1}\left[\dfrac{s - a}{(s - a)^2 + b^2}\right] = e^{at}\cos bt, L−1[b(s−a)2+b2]=eatsin⁡bt\mathcal{L}^{-1}\left[\dfrac{b}{(s - a)^2 + b^2}\right] = e^{at}\sin bt (learn this)