avewright/proofwiki-math huggingface dataset:
Theorem: Laplace Transform of t^3 cosh a t

:$\map {\laptrans {t^3 \cosh a t} } s = \dfrac {6 \paren {s^4 + 6 a^2 s^2 + a^4} } {\paren {s^2 - a^2}^4}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {t^3 \cosh a t} } s
      | r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t^2 \cosh a t} } s}
      | c = Derivative of Laplace Transform
}}
{{eqn | r = \map {\dfrac \d {\d s} } {-\dfrac {2 s^3 + 6 a^2 s} {\paren {s^2 - a^2}^3} }
      | c = Laplace Transform of $t^2 \cosh a t$
}}
{{eqn | r = \dfrac {\paren {s^2 - a^2}^3 \map {\dfrac \d {\d s} } {-\paren {2 s^3 + 6 a^2 s} } - \paren {-\paren {2 s^3 - 6 a^2 s} } \map {\dfrac \d {\d s} } {\paren {s^2 - a^2}^3} } {\paren {s^2 - a^2}^6}
      | c = Quotient Rule for Derivatives
}}
{{eqn | r = \dfrac {-\paren {s^2 - a^2}^3 \paren {6 a^2 + 6 s^2} + \paren {6 a^2 s + 2 s^3} \paren {3 \paren {s^2 - a^2}^2 \paren {2 s} } } {\paren {s^2 - a^2}^6}
      | c = Power Rule for Derivatives, Chain Rule for Derivatives
}}
{{eqn | r = \dfrac {6 \paren {a^2 - s^2} \paren {s^2 + a^2} + s \paren {6 a^2 s + 2 s^3} } {\paren {s^2 - a^2}^4}
      | c = simplifying
}}
{{eqn | r = \dfrac {6 \paren {a^4 - s^4 + 6 a^2 s^2 + 2 s^4} } {\paren {s^2 - a^2}^4}
      | c = multiplying out
}}
{{eqn | r = \dfrac {6 \paren {s^4 + 6 a^2 s^2 + a^4} } {\paren {s^2 - a^2}^4}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Category:Laplace Transforms involving Hyperbolic Cosine Function
Category:Inverse Laplace Transforms of Rational Functions
Category:Examples of Laplace Transforms
p3gntvwvs1sn1ma6aip95p7ywxz0res
Theorem: Laplace Transform of t^3 sinh a t over 24 a

:$\map {\laptrans {\dfrac {t^3 \sinh a t} {24 a} } } s = \dfrac {s^3 + a^2 s} {\paren {s^2 - a^2}^4}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^3 \sinh a t} {24 a} } } s
      | r = \dfrac 1 {24 a} \paren {\map {\laptrans {t^3 \sinh a t} } s}
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {24 a} \paren {\dfrac {24 a s \paren {s^2 + a^2} } {\paren {s^2 - a^2}^4} }
      | c = Laplace Transform of $t^3 \sinh a t$
}}
{{eqn | r = \dfrac {s^3 + a^2 s} {\paren {s^2 - a^2}^4}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Laplace Transform of t^3 sinh a t

:$\map {\laptrans {t^3 \sinh a t} } s = \dfrac {24 a s \paren {s^2 + a^2} } {\paren {s^2 - a^2}^4}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {t^3 \sinh a t} } s
      | r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t^2 \sinh a t} } s}
      | c = Derivative of Laplace Transform
}}
{{eqn | r = \map {\dfrac \d {\d s} } {-2 a \paren {\dfrac {3 s^2 + a^2} {\paren {s^2 - a^2}^3} } }
      | c = Laplace Transform of $t^2 \sinh a t$
}}
{{eqn | r = \map {\dfrac \d {\d s} } {\dfrac {-6 a s^2 - 2 a^3} {\paren {s^2 + a^2}^3} }
      | c = simplifying
}}
{{eqn | r = \dfrac {\paren {s^2 - a^2}^3 \map {\dfrac \d {\d s} } {-6 a s^2 - 2 a^3} - \paren {-6 a s^2 - 2 a^3} \map {\dfrac \d {\d s} } {\paren {s^2 - a^2}^3} } {\paren {s^2 - a^2}^6}
      | c = Quotient Rule for Derivatives
}}
{{eqn | r = \dfrac {\paren {s^2 - a^2}^3 \paren {-12 a s} + \paren {6 a s^2 + 2 a^3} \paren {3 \paren {s^2 - a^2}^2 \paren {2 s} } } {\paren {s^2 - a^2}^6}
      | c = Power Rule for Derivatives, Chain Rule for Derivatives
}}
{{eqn | r = \dfrac {6 \paren {-2 a s \paren {s^2 - a^2} + s \paren {6 a s^2 + 2 a^3} } } {\paren {s^2 + a^2}^4}
      | c = simplifying
}}
{{eqn | r = \dfrac {6 \paren {-2 a s^3 + 2 a^3 s + 2 a^3 s + 6 a s^3} } {\paren {s^2 + a^2}^4}
      | c = multiplying out
}}
{{eqn | r = \dfrac {24 a s \paren {s^2 + a^2} } {\paren {s^2 - a^2}^4}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Category:Laplace Transforms involving Hyperbolic Sine Function
Category:Inverse Laplace Transforms of Rational Functions
Category:Examples of Laplace Transforms
s075alozhqo2kylj124k11w12mfpp7j
Theorem: Inverse Laplace Transform of 1 over s^3 + a^3

:$\map {\laptrans {\dfrac {e^{a t / 2} } {3 a^2} \paren {\sqrt 3 \sin \dfrac {\sqrt 3} 2 a t - \cos \dfrac {\sqrt 3} 2 a t + e^{-3 a t / 2} } } } s = \dfrac 1 {s^3 + a^3}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {e^{a t / 2} } {3 a^2} \paren {\sqrt 3 \sin \dfrac {\sqrt 3} 2 a t - \cos \dfrac {\sqrt 3} 2 a t + e^{-3 a t / 2} } } } s
      | r = \dfrac {\sqrt 3} {3 a^2} \map {\laptrans {e^{a t / 2} \sin \dfrac {\sqrt 3} 2 a t} } s - \dfrac 1 {3 a^2} \map {\laptrans {e^{a t / 2} \cos \dfrac {\sqrt 3} 2 a t} } s + \dfrac 1 {3 a^2} \map {\laptrans {e^{a t / 2}  e^{-3 a t / 2} } } s
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac {\sqrt 3} {3 a^2} \paren {\dfrac {\sqrt 3 a / 2} {\paren {s - a / 2}^2 + \paren {\sqrt 3 a / 2}^2} } - \dfrac 1 {3 a^2} \map {\laptrans {e^{a t / 2} \cos \dfrac {\sqrt 3} 2 a t} } s + \dfrac 1 {3 a^2} \map {\laptrans {e^{-a t / 2} } } s
      | c = Laplace Transform of Exponential times Sine and simplification
}}
{{eqn | r = \dfrac {\sqrt 3} {3 a^2} \paren {\dfrac {\sqrt 3 a / 2} {\paren {s - a / 2}^2 + 3 a^2 / 4} } - \dfrac 1 {3 a^2} \paren {\dfrac {s - a / 2} {\paren {s - a / 2}^2 + \paren {\sqrt 3 a / 2}^2} } + \dfrac 1 {3 a^2} \map {\laptrans {e^{-a t / 2} } } s
      | c = Laplace Transform of Exponential times Cosine
}}
{{eqn | r = \dfrac {\sqrt 3} {3 a^2} \paren {\dfrac {\sqrt 3 a / 2} {\paren {s - a / 2}^2 + 3 a^2 / 4} } - \dfrac 1 {3 a^2} \paren {\dfrac {s - a / 2} {\paren {s - a / 2}^2 + 3 a^2 / 4} } + \dfrac 1 {3 a^2} \paren {\dfrac 1 {s + a} }
      | c = Laplace Transform of Exponential
}}
{{eqn | r = \dfrac 1 {3 a^2} \paren {\dfrac {3 a / 2 - s + a / 2} {\paren {s - a / 2}^2 + 3 a^2 / 4} + \dfrac 1 {s + a} }
      | c = simplification
}}
{{eqn | r = \dfrac 1 {3 a^2} \paren {\dfrac {2 a - s} {\paren {s^2 - a s + a^2 / 4} + 3 a^2 / 4} + \dfrac 1 {s + a} }
      | c = multiplying out square
}}
{{eqn | r = \dfrac 1 {3 a^2} \paren {\dfrac {2 a - s} {s^2 - a s + a^2} + \dfrac 1 {s + a} }
      | c = simplifying
}}
{{eqn | r = \dfrac 1 {3 a^2} \paren {\dfrac {\paren {2 a - s} \paren {a + s} + \paren {s^2 - a s + a^2} } {\paren {s^2 - a s + a^2} \paren {s + a} } }
      | c = common denominator
}}
{{eqn | r = \dfrac 1 {3 a^2} \paren {\dfrac {2 a^2 + a s - s^2 + s^2 - a s + a^2} {s^3 + a^3} }
      | c = multiplying out, and Sum of Two Cubes
}}
{{eqn | r = \dfrac 1 {s^3 + a^3}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Inverse Laplace Transform of s over s^3 + a^3

:$\map {\laptrans {\dfrac {e^{a t / 2} } {3 a} \paren {\cos \dfrac {\sqrt 3} 2 a t + \sqrt 3 \sin \dfrac {\sqrt 3} 2 a t - e^{-3 a t / 2} } } } s = \dfrac s {s^3 + a^3}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {e^{a t / 2} } {3 a} \paren {\cos \dfrac {\sqrt 3} 2 a t + \sqrt 3 \sin \dfrac {\sqrt 3} 2 a t - e^{-3 a t / 2} } } } s
      | r = \dfrac 1 {3 a} \map {\laptrans {e^{a t / 2} \cos \dfrac {\sqrt 3} 2 a t} } s + \dfrac {\sqrt 3} {3 a} \map {\laptrans {e^{a t / 2} \sin \dfrac {\sqrt 3} 2 a t} } s - \dfrac 1 {3 a} \map {\laptrans {e^{a t / 2}  e^{-3 a t / 2} } } s
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {3 a} \paren {\dfrac {s - a / 2} {\paren {s - a / 2}^2 + \paren {\sqrt 3 a / 2}^2} } + \dfrac {\sqrt 3}  {3 a} \map {\laptrans {e^{a t / 2} \sin \dfrac {\sqrt 3} 2 a t} } s - \dfrac 1 {3 a} \map {\laptrans {e^{-a t / 2} } } s
      | c = Laplace Transform of Exponential times Cosine and simplification
}}
{{eqn | r = \dfrac 1 {3 a} \paren {\dfrac {s - a / 2} {\paren {s - a / 2}^2 + 3 a^2 / 4} } + \dfrac {\sqrt 3} {3 a} \paren {\dfrac {\sqrt 3 a / 2} {\paren {s - a / 2}^2 + \paren {\sqrt 3 a / 2}^2} } - \dfrac 1 {3 a} \map {\laptrans {e^{-a t / 2} } } s
      | c = Laplace Transform of Exponential times Sine
}}
{{eqn | r = \dfrac 1 {3 a} \paren {\dfrac {s - a / 2} {\paren {s - a / 2}^2 + 3 a^2 / 4} } + \dfrac {\sqrt 3} {3 a} \paren {\dfrac {\sqrt 3 a / 2} {\paren {s - a / 2}^2 + 3 a^2 / 4} } - \dfrac 1 {3 a} \paren {\dfrac 1 {s + a} }
      | c = Laplace Transform of Exponential
}}
{{eqn | r = \dfrac 1 {3 a} \paren {\dfrac {s - a / 2 + 3 a / 2} {\paren {s - a / 2}^2 + 3 a^2 / 4} - \dfrac 1 {s + a} }
      | c = simplification
}}
{{eqn | r = \dfrac 1 {3 a} \paren {\dfrac {s + a} {\paren {s^2 - a s + a^2 / 4} + 3 a^2 / 4} - \dfrac 1 {s + a} }
      | c = multiplying out square
}}
{{eqn | r = \dfrac 1 {3 a} \paren {\dfrac {s + a} {s^2 - a s + a^2} - \dfrac 1 {s + a} }
      | c = simplifying
}}
{{eqn | r = \dfrac 1 {3 a} \paren {\dfrac {\paren {s + a}^2 - \paren {s^2 - a s + a^2} } {\paren {s^2 - a s + a^2} \paren {s + a} } }
      | c = common denominator
}}
{{eqn | r = \dfrac 1 {3 a} \paren {\dfrac {s^2 + 2 a s + a^2 - s^2 + a s - a^2} {s^3 + a^3} }
      | c = multiplying out, and Sum of Two Cubes
}}
{{eqn | r = \dfrac s {s^3 + a^3}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Inverse Laplace Transform of s^2 over s^3 + a^3

:$\map {\laptrans {\dfrac 1 3 \paren {e^{-a t} + 2 e^{a t / 2} \cos \dfrac {\sqrt 3 a t} 2} } } s = \dfrac {s^2} {s^3 + a^3}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac 1 3 \paren {e^{-a t} + 2 e^{a t / 2} \cos \dfrac {\sqrt 3 a t} 2} } } s 
      | r = \dfrac 1 3 \map {\laptrans {e^{-a t} } } s + \dfrac 2 3 \map {\laptrans {e^{a t / 2} \cos \dfrac {\sqrt 3 a t} 2} } s
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 3 \map {\laptrans {e^{-a t} } } s + \dfrac 2 3 \paren {\dfrac {s - a / 2} {\paren {s - a / 2}^2 + 3 a^2 / 4} }
      | c = Laplace Transform of Exponential times Cosine and simplification
}}
{{eqn | r = \dfrac 1 3 \paren {\dfrac 1 {s + a} + \dfrac {2 s - a} {\paren {s - a / 2}^2 + 3 a^2 / 4} }
      | c = Laplace Transform of Exponential and simplification
}}
{{eqn | r = \dfrac 1 3 \paren {\dfrac 1 {s + a} + \dfrac {2 s - a} {\paren {s^2 - a s + a^2 / 4} + 3 a^2 / 4} }
      | c = multiplying out square
}}
{{eqn | r = \dfrac 1 3 \paren {\dfrac 1 {s + a} + \dfrac {2 s - a} {s^2 - a s + a^2} }
      | c = simplifying
}}
{{eqn | r = \dfrac 1 3 \paren {\dfrac {\paren {s^2 - a s + a^2} + \paren {2 s - a} \paren {s + a} } {\paren {s^2 - a s + a^2} \paren {s + a} } }
      | c = common denominator
}}
{{eqn | r = \dfrac 1 3 \paren {\dfrac {s^2 - a s + a^2 + 2 s^2 - a s + 2 a s - a^2} {s^3 + a^3} }
      | c = multiplying out, and Sum of Two Cubes
}}
{{eqn | r = \dfrac {s^2} {s^3 + a^3}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Inverse Laplace Transform of 1 over s^3 - a^3

:$\map {\laptrans {\dfrac {e^{-a t / 2} } {3 a^2} \paren {e^{3 a t / 2} - \cos \dfrac {\sqrt 3} 2 a t - \sqrt 3 \sin \dfrac {\sqrt 3} 2 a t} } } s = \dfrac 1 {s^3 - a^3}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {e^{-a t / 2} } {3 a^2} \paren {e^{3 a t / 2} - \cos \dfrac {\sqrt 3} 2 a t - \sqrt 3 \sin \dfrac {\sqrt 3} 2 a t} } } s
      | r = \dfrac 1 {3 a^2} \map {\laptrans {e^{a t} } } s - \dfrac 1 {3 a^2} \map {\laptrans {e^{-a t / 2} \cos \dfrac {\sqrt 3} 2 a t} } s - \dfrac {\sqrt 3} {3 a^2} \map {\laptrans {-e^{a t / 2} \sin \dfrac {\sqrt 3} 2 a t} } s
      | c = Linear Combination of Laplace Transforms and simplification
}}
{{eqn | r = \dfrac 1 {3 a^2} \paren {\dfrac 1 {s - a} } - \dfrac 1 {3 a^2} \map {\laptrans {e^{-a t / 2} \cos \dfrac {\sqrt 3} 2 a t} } s - \dfrac {\sqrt 3} {3 a^2} \map {\laptrans {-e^{a t / 2} \sin \dfrac {\sqrt 3} 2 a t} } s
      | c = Laplace Transform of Exponential
}}
{{eqn | r = \dfrac 1 {3 a^2} \paren {\dfrac 1 {s - a} } - \dfrac 1 {3 a^2} \paren {\dfrac {s + a / 2} {\paren {s + a / 2}^2 + \paren {\sqrt 3 a / 2}^2} } - \dfrac {\sqrt 3} {3 a^2} \map {\laptrans {-e^{a t / 2} \sin \dfrac {\sqrt 3} 2 a t} } s
      | c = Laplace Transform of Exponential times Cosine
}}
{{eqn | r = \dfrac 1 {3 a^2} \paren {\dfrac 1 {s - a} } - \dfrac 1 {3 a^2} \paren {\dfrac {s + a / 2} {\paren {s + a / 2}^2 + 3 a^2 / 4} } - \dfrac 1 {3 a^2} \paren {\dfrac {3 a / 2} {\paren {s + a / 2}^2 + 3 a^2 / 4} }
      | c = Laplace Transform of Exponential times Sine and simplification
}}
{{eqn | r = \dfrac 1 {3 a^2} \paren {\dfrac 1 {s - a} - \dfrac {s + a / 2 + 3 a / 2} {\paren {s + a / 2}^2 + 3 a^2 / 4} }
      | c = simplification
}}
{{eqn | r = \dfrac 1 {3 a^2} \paren {\dfrac 1 {s - a} - \dfrac {s + 2 a} {s^2 + a s + a^2} }
      | c = multiplying out square and simplifying
}}
{{eqn | r = \dfrac 1 {3 a^2} \paren {\dfrac {\paren {s^2 + a s + a^2} - \paren {s + 2 a} \paren {s - a} } {\paren {s - a} \paren {s^2 + a s + a^2} } }
      | c = common denominator
}}
{{eqn | r = \dfrac 1 {3 a^2} \paren {\dfrac {s^2 + a s + a^2 - s^2 - 2 a s + a s + 2 a^2} {s^3 - a^3} }
      | c = multiplying out, and Difference of Two Cubes
}}
{{eqn | r = \dfrac 1 {s^3 - a^3}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Inverse Laplace Transform of s over s^3 - a^3

:$\map {\laptrans {\dfrac {e^{-a t / 2} } {3 a} \paren {\sqrt 3 \sin \dfrac {\sqrt 3} 2 a t - \cos \dfrac {\sqrt 3} 2 a t + e^{3 a t / 2} } } } s = \dfrac s {s^3 - a^3}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {e^{-a t / 2} } {3 a} \paren {\sqrt 3 \sin \dfrac {\sqrt 3} 2 a t - \cos \dfrac {\sqrt 3} 2 a t + e^{3 a t / 2} } } } s
      | r = \dfrac {\sqrt 3} {3 a} \map {\laptrans {e^{-a t / 2} \sin \dfrac {\sqrt 3} 2 a t} } s - \dfrac 1 {3 a} \map {\laptrans {e^{-a t / 2} \cos \dfrac {\sqrt 3} 2 a t} } s + \dfrac 1 {3 a} \map {\laptrans {e^{-a t / 2} e^{3 a t / 2} } } s
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac {\sqrt 3}  {3 a} \map {\laptrans {e^{-a t / 2} \sin \dfrac {\sqrt 3} 2 a t} } s - \dfrac 1 {3 a} \paren {\dfrac {s + a / 2} {\paren {s + a / 2}^2 + \paren {\sqrt 3 a / 2}^2} } + \dfrac 1 {3 a} \map {\laptrans {e^{a t / 2} } } s
      | c = Laplace Transform of Exponential times Cosine and simplification
}}
{{eqn | r = \dfrac {\sqrt 3} {3 a} \paren {\dfrac {\sqrt 3 a / 2} {\paren {s + a / 2}^2 + \paren {\sqrt 3 a / 2}^2} } - \dfrac 1 {3 a} \paren {\dfrac {s + a / 2} {\paren {s + a / 2}^2 + 3 a^2 / 4} } + \dfrac 1 {3 a} \map {\laptrans {e^{a t / 2} } } s
      | c = Laplace Transform of Exponential times Sine
}}
{{eqn | r = \dfrac {\sqrt 3} {3 a} \paren {\dfrac {\sqrt 3 a / 2} {\paren {s + a / 2}^2 + 3 a^2 / 4} } - \dfrac 1 {3 a} \paren {\dfrac {s + a / 2} {\paren {s + a / 2}^2 + 3 a^2 / 4} } + \dfrac 1 {3 a} \paren {\dfrac 1 {s - a} }
      | c = Laplace Transform of Exponential
}}
{{eqn | r = \dfrac 1 {3 a} \paren {\dfrac {3 a / 2 - s - a / 2} {\paren {s + a / 2}^2 + 3 a^2 / 4} + \dfrac 1 {s - a} }
      | c = simplification
}}
{{eqn | r = \dfrac 1 {3 a} \paren {\dfrac {a - s} {\paren {s^2 + a s + a^2 / 4} + 3 a^2 / 4} + \dfrac 1 {s - a} }
      | c = multiplying out square
}}
{{eqn | r = \dfrac 1 {3 a} \paren {\dfrac {a - s} {s^2 + a s + a^2} + \dfrac 1 {s - a} }
      | c = simplifying
}}
{{eqn | r = \dfrac 1 {3 a} \paren {\dfrac {\paren {s^2 + a s + a^2} - \paren {s - a}^2} {\paren {s^2 + a s + a^2} \paren {s - a} } }
      | c = common denominator
}}
{{eqn | r = \dfrac 1 {3 a} \paren {\dfrac {s^2 + a s + a^2 - s^2 + 2 a s - a^2} {s^3 - a^3} }
      | c = multiplying out, and Difference of Two Cubes
}}
{{eqn | r = \dfrac s {s^3 - a^3}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Inverse Laplace Transform of s^2 over s^3 - a^3

:$\map {\laptrans {\dfrac 1 3 \paren {e^{a t} + 2 e^{-a t / 2} \cos \dfrac {\sqrt 3 a t} 2} } } s = \dfrac {s^2} {s^3 - a^3}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac 1 3 \paren {e^{a t} + 2 e^{-a t / 2} \cos \dfrac {\sqrt 3 a t} 2} } } s 
      | r = \dfrac 1 3 \map {\laptrans {e^{a t} } } s + \dfrac 2 3 \map {\laptrans {e^{-a t / 2} \cos \dfrac {\sqrt 3 a t} 2} } s
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 3 \map {\laptrans {e^{a t} } } s + \dfrac 2 3 \paren {\dfrac {s + a / 2} {\paren {s + a / 2}^2 + 3 a^2 / 4} }
      | c = Laplace Transform of Exponential times Cosine and simplification
}}
{{eqn | r = \dfrac 1 3 \paren {\dfrac 1 {s - a} + \dfrac {2 s + a} {\paren {s + a / 2}^2 + 3 a^2 / 4} }
      | c = Laplace Transform of Exponential and simplification
}}
{{eqn | r = \dfrac 1 3 \paren {\dfrac 1 {s - a} + \dfrac {2 s + a} {\paren {s^2 + a s + a^2 / 4} + 3 a^2 / 4} }
      | c = multiplying out square
}}
{{eqn | r = \dfrac 1 3 \paren {\dfrac 1 {s - a} + \dfrac {2 s + a} {s^2 + a s + a^2} }
      | c = simplifying
}}
{{eqn | r = \dfrac 1 3 \paren {\dfrac {\paren {s^2 + a s + a^2} + \paren {2 s + a} \paren {s - a} } {\paren {s^2 + a s + a^2} \paren {s - a} } }
      | c = common denominator
}}
{{eqn | r = \dfrac 1 3 \paren {\dfrac {s^2 + a s + a^2 + 2 s^2 + a s - 2 a s - a^2} {s^3 - a^3} }
      | c = multiplying out, and Difference of Two Cubes
}}
{{eqn | r = \dfrac {s^2} {s^3 - a^3}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Inverse Laplace Transform of 1 over s^4 + 4 a^4

:$\map {\laptrans {\dfrac 1 {4 a^3} \paren {\sin a t \cosh a t - \cos a t \sinh a t} } } s = \dfrac 1 {s^4 + 4 a^4}$

Proof:
First we note that:
{{begin-eqn}}
{{eqn | l = \sin a t \cosh a t - \cos a t \sinh a t
      | r = \sin a t \paren {\dfrac {e^{a t} + e^{-a t} } 2} - \cos a t \paren {\dfrac {e^{a t} - e^{-a t} } 2}
      | c = {{Defof|Hyperbolic Cosine}} and {{Defof|Hyperbolic Sine}}
}}
{{eqn | n = 1
      | r = \dfrac 1 2 \paren {e^{a t} \sin a t + e^{-a t} \sin a t - e^{a t} \cos a t + e^{-a t} \cos a t}
      | c = 
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac 1 {4 a^3} \paren {\sin a t \cosh a t - \cos a t \sinh a t} } } s
      | r = \map {\laptrans {\dfrac 1 {8 a^3} \paren {e^{a t} \sin a t + e^{-a t} \sin a t - e^{a t} \cos a t + e^{-a t} \cos a t} } } s
      | c = from $(1)$
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\map {\laptrans {e^{a t} \sin a t} } s + \map {\laptrans {e^{-a t} \sin a t} } s - \map {\laptrans {e^{a t} \cos a t} } s + \map {\laptrans {e^{-a t} \cos a t} } s}
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac a {\paren {s - a}^2 + a^2} + \dfrac a {\paren {s + a}^2 + a^2} - \map {\laptrans {e^{a t} \cos a t} } s + \map {\laptrans {e^{-a t} \cos a t} } s}
      | c = Laplace Transform of Exponential times Sine
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac a {\paren {s - a}^2 + a^2} + \dfrac a {\paren {s + a}^2 + a^2} - \dfrac {s - a} {\paren {s - a}^2 + a^2} + \dfrac {s + a} {\paren {s + a}^2 + a^2} }
      | c = Laplace Transform of Exponential times Cosine
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac {2 a - s} {\paren {s - a}^2 + a^2} + \dfrac {2 a + s} {\paren {s + a}^2 + a^2} }
      | c = simplifying
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac {\paren {2 a - s} \paren {\paren {s + a}^2 + a^2} + \paren {2 a + s} \paren {\paren {s - a}^2 + a^2} } {\paren {\paren {s - a}^2 + a^2} \paren {\paren {s + a}^2 + a^2} } }
      | c = common denominator
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac {\paren {2 a - s} \paren {s^2 + 2 a s + 2 a^2} + \paren {2 a + s} \paren {s^2 - 2 a s + 2 a^2} } {\paren {s^2 - 2 a s + 2 a^2} \paren {s^2 + 2 a s + 2 a^2} } }
      | c = multiplying out
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac {2 a s^2 + 4 a^2 s + 4 a^3 - s^3 - 2 a s^2 - 2 a^2 s + 2 a s^2 - 4 a^2 s + 4 a^3 + s^3 - 2 a s^2 + 2 a^2 s} {s^4 + 4 a^4} }
      | c = multiplying out, and Sophie Germain's Identity
}}
{{eqn | r = \dfrac 1 {s^4 + 4 a^4}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Inverse Laplace Transform of s over s^4 + 4 a^4

:$\map {\laptrans {\dfrac {\sin a t \sinh a t} {2 a^2} } } s = \dfrac s {s^4 + 4 a^4}$

Proof:
First we note that:
{{begin-eqn}}
{{eqn | l = \sin a t \sinh a t
      | r = \sin a t \paren {\dfrac {e^{a t} - e^{-a t} } 2}
      | c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | n = 1
      | r = \dfrac 1 2 \paren {e^{a t} \sin a t - e^{-a t} \sin a t}
      | c = 
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\sin a t \sinh a t} {2 a^2} } } s
      | r = \map {\laptrans {\dfrac 1 {4 a^2} \paren {e^{a t} \sin a t - e^{-a t} \sin a t} } } s
      | c = from $(1)$
}}
{{eqn | r = \dfrac 1 {4 a^2} \paren {\map {\laptrans {e^{a t} \sin a t} } s - \map {\laptrans {e^{-a t} \sin a t} } s}
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {4 a^2} \paren {\dfrac a {\paren {s - a}^2 + a^2} - \dfrac a {\paren {s + a}^2 + a^2} }
      | c = Laplace Transform of Exponential times Sine
}}
{{eqn | r = \dfrac 1 {4 a^2} \paren {\dfrac {a \paren {\paren {s + a}^2 + a^2} - a \paren {\paren {s - a}^2 + a^2} } {\paren {\paren {s - a}^2 + a^2} \paren {\paren {s + a}^2 + a^2} } }
      | c = common denominator
}}
{{eqn | r = \dfrac 1 {4 a^2} \paren {\dfrac {a \paren {s^2 + 2 a s + 2 a^2} - a \paren {s^2 - 2 a s + 2 a^2} } {\paren {s^2 - 2 a s + 2 a^2} \paren {s^2 + 2 a s + 2 a^2} } }
      | c = multiplying out
}}
{{eqn | r = \dfrac 1 {4 a^2} \paren {\dfrac {a s^2 + 2 a^2 s + 2 a^3 - a s^2 + 2 a^2 s - 2 a^3} {s^4 + 4 a^4} }
      | c = multiplying out, and Sophie Germain's Identity
}}
{{eqn | r = \dfrac s {s^4 + 4 a^4}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Inverse Laplace Transform of s^2 over s^4 + 4 a^4

:$\map {\laptrans {\dfrac 1 {2 a} \paren {\sin a t \cosh a t + \cos a t \sinh a t} } } s = \dfrac {s^2} {s^4 + 4 a^4}$

Proof:
First we note that:
{{begin-eqn}}
{{eqn | l = \sin a t \cosh a t + \cos a t \sinh a t
      | r = \sin a t \paren {\dfrac {e^{a t} + e^{-a t} } 2} + \cos a t \paren {\dfrac {e^{a t} - e^{-a t} } 2}
      | c = {{Defof|Hyperbolic Cosine}} and {{Defof|Hyperbolic Sine}}
}}
{{eqn | n = 1
      | r = \dfrac 1 2 \paren {e^{a t} \sin a t + e^{-a t} \sin a t + e^{a t} \cos a t - e^{-a t} \cos a t}
      | c = 
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac 1 {2 a} \paren {\sin a t \cosh a t + \cos a t \sinh a t} } } s
      | r = \map {\laptrans {\dfrac 1 {4 a} \paren {e^{a t} \sin a t + e^{-a t} \sin a t + e^{a t} \cos a t - e^{-a t} \cos a t} } } s
      | c = from $(1)$
}}
{{eqn | r = \dfrac 1 {4 a} \paren {\map {\laptrans {e^{a t} \sin a t} } s + \map {\laptrans {e^{-a t} \sin a t} } s + \map {\laptrans {e^{a t} \cos a t} } s - \map {\laptrans {e^{-a t} \cos a t} } s}
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {4 a} \paren {\dfrac a {\paren {s - a}^2 + a^2} + \dfrac a {\paren {s + a}^2 + a^2} + \map {\laptrans {e^{a t} \cos a t} } s - \map {\laptrans {e^{-a t} \cos a t} } s}
      | c = Laplace Transform of Exponential times Sine
}}
{{eqn | r = \dfrac 1 {4 a} \paren {\dfrac a {\paren {s - a}^2 + a^2} + \dfrac a {\paren {s + a}^2 + a^2} + \dfrac {s - a} {\paren {s - a}^2 + a^2} - \dfrac {s + a} {\paren {s + a}^2 + a^2} }
      | c = Laplace Transform of Exponential times Cosine
}}
{{eqn | r = \dfrac 1 {4 a} \paren {\dfrac s {\paren {s - a}^2 + a^2} - \dfrac s {\paren {s + a}^2 + a^2} }
      | c = simplifying
}}
{{eqn | r = \dfrac 1 {4 a} \paren {\dfrac {s \paren {\paren {s + a}^2 + a^2} - s \paren {\paren {s - a}^2 + a^2} } {\paren {\paren {s - a}^2 + a^2} \paren {\paren {s + a}^2 + a^2} } }
      | c = common denominator
}}
{{eqn | r = \dfrac 1 {4 a} \paren {\dfrac {s \paren {s^2 + 2 a s + 2 a^2} - s \paren {s^2 - 2 a s + 2 a^2} } {\paren {s^2 - 2 a s + 2 a^2} \paren {s^2 + 2 a s + 2 a^2} } }
      | c = multiplying out
}}
{{eqn | r = \dfrac 1 {4 a} \paren {\dfrac {s^3 + 2 a s^2 + 2 a^2 s - s^3 + 2 a s^2 - 2 a^2 s} {s^4 + 4 a^4} }
      | c = multiplying out, and Sophie Germain's Identity
}}
{{eqn | r = \dfrac {s^2} {s^4 + 4 a^4}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}