function f(t) 1 tn eat sin at cos at sinh at cosh at e-at f(t) U(ta) or Ua(t) (a ≥ 0) (ta) (a> 0) U(t − a)f(t − a) or Ua(t)f(t − a) f(n)(t) (−t)" f (t) - (fg)(t) = f(r)g(t − T) dr Laplace transform F(s) 1/s (s > 0) n!/sn+1 (s > 0) 1/(s – a) (s> a) a/(s²+a²) (s > 0) s/(s² + a²) (s > 0) a/(s² - a²) (s> |a|) s/(s² – a²) F(s+a (s>|a|) -as e S S 0) as e as e S F(s F(n)(s) F(s)G(s) sn ¯¹ƒ (0) — sn−² ƒ'(0) … … … — ƒ (n−¹) (0) Solve the differential equation บ'' "-y-12y=-4-5₁(t), y(0) = −2, y'(0) = 1 using Laplace transforms. The solution is y(t) and y(t): = = fort > 4 for 0 < t <4

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