avewright/proofwiki-math huggingface dataset:
Theorem: Inverse Laplace Transform of s^5 over (s^2 + a^2)^3

:$\map {\laptrans {\dfrac {\paren {8 - a^2 t^2} \cos a t - 7 a t \sin a t} 8} } s = \dfrac {s^5} {\paren {s^2 + a^2}^3}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {8 - a^2 t^2} \cos a t - 7 a t \sin a t} 8} } s
      | r = \dfrac 1 8 \paren {8 \map {\laptrans {\cos a t} } s - a^2 \map {\laptrans {t^2 \cos a t} } s - 7 a \map {\laptrans {t \sin a t} } s}
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 8 \paren {8 \dfrac s {s^2 + a^2} - a^2 \map {\laptrans {t^2 \cos a t} } s - 7 a \map {\laptrans {t \sin a t} } s}
      | c = Laplace Transform of Cosine
}}
{{eqn | r = \dfrac 1 8 \paren {\dfrac {8 s} {s^2 + a^2} - a^2 \map {\laptrans {t^2 \cos a t} } s - 7 a \dfrac {2 a s} {\paren {s^2 + a^2}^2} }
      | c = Laplace Transform of $t \sin a t$
}}
{{eqn | r = \dfrac 1 8 \paren {\dfrac {8 s} {s^2 + a^2} - a^2 \paren {\dfrac {2 s \paren {s^2 - 3 a^2} } {\paren {s^2 + a^2}^3} } - \dfrac {14 a^2 s} {\paren {s^2 + a^2}^2} }
      | c = Laplace Transform of $t^2 \cos a t$
}}
{{eqn | r = \dfrac 1 8 \paren {\dfrac {8 s} {s^2 + a^2} - \dfrac {2 a^2 s \paren {s^2 - 3 a^2} } {\paren {s^2 + a^2}^3} - \dfrac {14 a^2 s} {\paren {s^2 + a^2}^2} }
      | c = simplifying
}}
{{eqn | r = \dfrac 1 8 \paren {\dfrac {8 s \paren {s^2 + a^2}^2 - 2 a^2 s \paren {s^2 - 3 a^2} - 14 a^2 s \paren {s^2 + a^2} } {\paren {s^2 + a^2}^3} }
      | c = common denominator
}}
{{eqn | r = \dfrac 1 8 \paren {\dfrac {8 s^5 + 16 a^2 s^3 + 8 a^4 s - 2 a^2 s^3 + 6 a^4 s - 14 a^2 s^3 - 14 a^4 s} {\paren {s^2 + a^2}^3} }
      | c = multiplying out
}}
{{eqn | r = \dfrac 1 8 \paren {\dfrac {8 s^5} {\paren {s^2 + a^2}^3} }
      | c = simplifying
}}
{{eqn | r = \dfrac {s^5} {\paren {s^2 + a^2}^3}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Laplace Transform of t^2 sine a t over 2 a

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

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

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

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

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

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {t^3 \cos a t} } s
      | r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t^2 \cos 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 $t^2 \cos a t$
}}
{{eqn | r = \map {\dfrac \d {\d s} } {\dfrac {6 a^2 s - 2 s^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^2 s - 2 s^3} - \paren {6 a^2 s - 2 s^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 {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 {\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 {-s^4 + a^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 Cosine Function
Category:Inverse Laplace Transforms of Rational Functions
Category:Examples of Laplace Transforms
9yixw92jt1fyxwxidvdtetu9ohshhj2
Theorem: Laplace Transform of t^3 cosine a t over 6

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

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^3 \cos a t} 6} } s
      | r = \dfrac 1 6 \paren {\map {\laptrans {t^3 \cos a t} } s}
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 6 \paren {\dfrac {6 \paren {s^4 - 6 a^2 s^2 + a^4} } {\paren {s^2 + a^2}^4} }
      | c = Laplace Transform of $t^3 \cos a t$
}}
{{eqn | r = \dfrac {s^4 - 6 a^2 s^2 + a^4} {\paren {s^2 + a^2}^4}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Laplace Transform of t^3 sine a t over 24 a

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

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^3 \sin a t} {24 a} } } s
      | r = \dfrac 1 {24 a} \paren {\map {\laptrans {t^3 \sin 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 $t^3 \sin a t$
}}
{{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 sine a t

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

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {t^3 \sin a t} } s
      | r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t^2 \sin 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 $t^2 \sin a t$
}}
{{eqn | r = \map {\dfrac \d {\d s} } {\dfrac {2 a^3 - 6 a s^2} {\paren {s^2 + a^2}^3} }
      | c = simplifying
}}
{{eqn | r = \dfrac {\paren {s^2 + a^2}^3 \map {\dfrac \d {\d s} } {2 a^3 - 6 a s^2} - \paren {2 a^3 - 6 a s^2} \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 {2 a^3 - 6 a s^2} \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 {2 a^3 - 6 a s^2} } } {\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 Sine Function
Category:Inverse Laplace Transforms of Rational Functions
Category:Examples of Laplace Transforms
cd9z7b1jzs6jqn2rpjs9ef84wbb21sq
Theorem: Inverse Laplace Transform of 1 over (s^2 - a^2)^3

:$\map {\laptrans {\dfrac {\paren {3 + a^2 t^2} \sinh a t - 3 a t \cosh a t} {8 a^5} } } s = \dfrac 1 {\paren {s^2 - a^2}^3}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {3 + a^2 t^2} \sinh a t - 3 a t \cosh a t} {8 a^5} } } s
      | r = \dfrac 1 {8 a^5} \paren {3 \map {\laptrans {\sinh a t} } s + a^2 \map {\laptrans {t^2 \sinh a t} } s - 3 a \map {\laptrans {t \cosh a t} } s}
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {8 a^5} \paren {\dfrac {3 a} {s^2 - a^2} + a^2 \map {\laptrans {t^2 \sinh a t} } s - 3 a \map {\laptrans {t \cosh a t} } s}
      | c = Laplace Transform of Hyperbolic Sine
}}
{{eqn | r = \dfrac 1 {8 a^5} \paren {\dfrac {3 a} {s^2 - a^2} + a^2 \map {\laptrans {t^2 \sinh a t} } s - \dfrac {3 a \paren {s^2 + a^2} } {\paren {s^2 - a^2}^2} }
      | c = Laplace Transform of $t \cosh a t$
}}
{{eqn | r = \dfrac 1 {8 a^5} \paren {\dfrac {3 a} {s^2 - a^2} + a^2 \paren {2 a \paren {\dfrac {3 s^2 + a^2} {\paren {s^2 - a^2}^3} } } - \dfrac {3 a \paren {s^2 + a^2} } {\paren {s^2 - a^2}^2} }
      | c = Laplace Transform of $t^2 \sinh a t$
}}
{{eqn | r = \dfrac 1 {8 a^5} \paren {\dfrac {3 a} {s^2 - a^2} + \dfrac {2 a^3 \paren {3 s^2 + a^2} } {\paren {s^2 - a^2}^3} - \dfrac {3 a \paren {s^2 + a^2} } {\paren {s^2 - a^2}^2} }
      | c = simplifying
}}
{{eqn | r = \dfrac 1 {8 a^5} \paren {\dfrac {3 a \paren {s^2 - a^2}^2 + 2 a^3 \paren {3 s^2 + a^2} - 3 a \paren {s^2 - a^2} \paren {s^2 + a^2} } {\paren {s^2 - a^2}^3} }
      | c = common denominator
}}
{{eqn | r = \dfrac 1 {8 a^5} \paren {\dfrac {3 a s^4 - 6 a^3 s^2 + 3 a^5 + 6 a^3 s^2 + 2 a^5 - 3 a s^4 + 3 a^5} {\paren {s^2 - a^2}^3} }
      | c = multiplying out
}}
{{eqn | r = \dfrac 1 {8 a^5} \paren {\dfrac {8 a^5} {\paren {s^2 - a^2}^3} }
      | c = simplifying
}}
{{eqn | r = \dfrac 1 {\paren {s^2 - a^2}^3}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Inverse Laplace Transform of s over (s^2 - a^2)^3

:$\map {\laptrans {\dfrac {a t^2 \cosh a t - t \sinh a t} {8 a^3} } } s = \dfrac s {\paren {s^2 - a^2}^3}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {a t^2 \cosh a t - t \sinh a t} {8 a^3} } } s
      | r = \dfrac 1 {8 a^3} \paren {a \map {\laptrans {t^2 \cosh a t}  - \map {\laptrans {t \sinh a t} } s} s}
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {a \map {\laptrans {t^2 \cosh a t} } s - \dfrac {2 a s} {\paren {s^2 - a^2}^2} }
      | c = Laplace Transform of $t \sinh a t$
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac {2 a s \paren {s^2 + 3 a^2} } {\paren {s^2 - a^2}^3} - \dfrac {2 a s} {\paren {s^2 - a^2}^2} }
      | c = Laplace Transform of $t^2 \cosh a t$
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac {2 a s \paren {s^2 + 3 a^2} - 2 a s \paren {s^2 - a^2} } {\paren {s^2 - a^2}^3} }
      | c = common denominator
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac {2 a s^3 + 6 a^3 s - 2 a s^3 + 2 a^3 s} {\paren {s^2 - a^2}^3} }
      | c = multiplying out
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac {8 a^3 s} {\paren {s^2 - a^2}^3} }
      | c = simplifying
}}
{{eqn | r = \dfrac s {\paren {s^2 - a^2}^3}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Laplace Transform of t^2 cosh a t

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

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {t^2 \cosh a t} } s
      | r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t \cosh a t} } s}
      | c = Derivative of Laplace Transform
}}
{{eqn | r = \map {\dfrac \d {\d s} } {-\dfrac {s^2 + a^2} {\paren {s^2 - a^2}^2} }
      | c = Laplace Transform of $t \cosh a t$
}}
{{eqn | r = \dfrac {-\paren {s^2 - a^2}^2 \map {\dfrac \d {\d s} } {s^2 + a^2} + \paren {s^2 + a^2} \map {\dfrac \d {\d s} } {\paren {s^2 - a^2}^2} } {\paren {s^2 - a^2}^4}
      | c = Quotient Rule for Derivatives
}}
{{eqn | r = \dfrac {-\paren {s^2 - a^2}^2 \paren {2 s} + \paren {s^2 + a^2} \paren {2 \paren {s^2 - a^2} \paren {2 s} } } {\paren {s^2 - a^2}^4}
      | c = Power Rule for Derivatives, Chain Rule for Derivatives
}}
{{eqn | r = \dfrac {-2 s \paren {s^2 - a^2} + 4 s \paren {s^2 + a^2} } {\paren {s^2 - a^2}^3}
      | c = simplifying
}}
{{eqn | r = \dfrac {-2 s^3 + 2 s a^2 + 4 s a^2 + 4 s^3} {\paren {s^2 - a^2}^3}
      | c = multiplying out
}}
{{eqn | r = \dfrac {2 s^3 + 6 a^2 s} {\paren {s^2 - a^2}^3}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Category:Laplace Transforms involving Hyperbolic Cosine Function
Category:Inverse Laplace Transforms of Rational Functions
Category:Examples of Laplace Transforms
idybo3ctyo6n3w30dgzbs5yhonr7bmk
Theorem: Inverse Laplace Transform of s^2 over (s^2 - a^2)^3

:$\map {\laptrans {\dfrac {\paren {a^2 t^2 - 1} \sinh a t + a t \cosh a t} {8 a^3} } } s = \dfrac {s^2} {\paren {s^2 - a^2}^3}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {a^2 t^2 - 1} \sinh a t + a t \cosh a t} {8 a^3} } } s
      | r = \dfrac 1 {8 a^3} \paren {a^2 \map {\laptrans {t^2 \sinh a t} } s - \map {\laptrans {\sinh a t} } s + a \map {\laptrans {t \cosh a t} } s}
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {a^2 \map {\laptrans {t^2 \sin a t} } s - \dfrac a {s^2 - a^2} + a \map {\laptrans {t \cos a t} } s}
      | c = Laplace Transform of Hyperbolic Sine
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {a^2 \map {\laptrans {t^2 \sin a t} } s - \dfrac a {s^2 - a^2} + \dfrac {a \paren {s^2 + a^2} } {\paren {s^2 - a^2}^2} }
      | c = Laplace Transform of $t \cosh a t$
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {a^2 \paren {2 a \paren {\dfrac {3 s^2 + a^2} {\paren {s^2 - a^2}^3} } } - \dfrac a {s^2 - a^2} + \dfrac {a \paren {s^2 + a^2} } {\paren {s^2 - a^2}^2} }
      | c = Laplace Transform of $t^2 \sinh a t$
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac {2 a^3 \paren {3 s^2 + a^2} } {\paren {s^2 - a^2}^3} - \dfrac a {s^2 - a^2} + \dfrac {a \paren {s^2 + a^2} } {\paren {s^2 - a^2}^2} }
      | c = simplifying
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac {2 a^3 \paren {3 s^2 + a^2} - a \paren {s^2 - a^2}^2 + a \paren {s^2 + a^2} \paren {s^2 - a^2} } {\paren {s^2 - a^2}^3} }
      | c = common denominator
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac {6 a^3 s^2 + 2 a^5 - a s^4 + 2 a^3 s^2 - a^5 + a s^4 - a^5} {\paren {s^2 - a^2}^3} }
      | c = multiplying out
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac {8 a^3 s^2} {\paren {s^2 - a^2}^3} }
      | c = simplifying
}}
{{eqn | r = \dfrac {s^2} {\paren {s^2 - a^2}^3}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Inverse Laplace Transform of s^3 over (s^2 - a^2)^3

:$\map {\laptrans {\dfrac {3 t \sinh a t + a t^2 \cosh a t} {8 a} } } s = \dfrac {s^3} {\paren {s^2 - a^2}^3}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {3 t \sinh a t + a t^2 \cosh a t} {8 a} } } s
      | r = \dfrac 1 {8 a} \paren {3 \map {\laptrans {t \sinh a t} } s + a \map {\laptrans {t^2 \cosh a t} } s}
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {8 a} \paren {3 \dfrac {2 a s} {\paren {s^2 - a^2}^2} + a \map {\laptrans {t^2 \cosh a t} } s}
      | c = Laplace Transform of $t \sinh a t$
}}
{{eqn | r = \dfrac 1 {8 a} \paren {\dfrac {6 a s} {\paren {s^2 - a^2}^2} + \dfrac {2 a s \paren {s^2 + 3 a^2} } {\paren {s^2 - a^2}^3} }
      | c = Laplace Transform of $t^2 \cosh a t$
}}
{{eqn | r = \dfrac 1 {8 a} \paren {\dfrac {6 a s \paren {s^2 - a^2} + 2 a s \paren {s^2 + 3 a^2} } {\paren {s^2 - a^2}^3} }
      | c = common denominator
}}
{{eqn | r = \dfrac 1 {8 a} \paren {\dfrac {6 a s^3 - 6 a^3 s + 2 a s^3 + 6 a^3 s} {\paren {s^2 - a^2}^3} }
      | c = multiplying out
}}
{{eqn | r = \dfrac 1 {8 a} \paren {\dfrac {8 a s^3} {\paren {s^2 - a^2}^3} }
      | c = simplifying
}}
{{eqn | r = \dfrac {s^3} {\paren {s^2 - a^2}^3}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Inverse Laplace Transform of s^4 over (s^2 - a^2)^3

:$\map {\laptrans {\dfrac {\paren {3 + a^2 t^2} \sinh a t + 5 a t \cosh a t} {8 a} } } s = \dfrac {s^4} {\paren {s^2 - a^2}^3}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {3 + a^2 t^2} \sinh a t + 5 a t \cosh a t} {8 a} } } s
      | r = \dfrac 1 {8 a} \paren {3 \map {\laptrans {\sinh a t} } s + a^2 \map {\laptrans {t^2 \sinh a t} } s + 5 a \map {\laptrans {t \cosh a t} } s}
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {8 a} \paren {3 \dfrac a {s^2 - a^2} + a^2 \map {\laptrans {t^2 \sinh a t} } s + 5 a \map {\laptrans {t \cosh a t} } s}
      | c = Laplace Transform of Hyperbolic Sine
}}
{{eqn | r = \dfrac 1 {8 a} \paren {\dfrac {3 a} {s^2 - a^2} + a^2 \map {\laptrans {t^2 \sinh a t} } s + 5 \dfrac {a \paren {s^2 + a^2} } {\paren {s^2 - a^2}^2} }
      | c = Laplace Transform of $t \cosh a t$
}}
{{eqn | r = \dfrac 1 {8 a} \paren {\dfrac {3 a} {s^2 - a^2} + a^2 \paren {2 a \paren {\dfrac {3 s^2 + a^2} {\paren {s^2 + a^2}^3} } } + \dfrac {5 a \paren {s^2 + a^2} } {\paren {s^2 - a^2}^2} }
      | c = Laplace Transform of $t^2 \sinh a t$
}}
{{eqn | r = \dfrac 1 {8 a} \paren {\dfrac {3 a} {s^2 - a^2} + \dfrac {2 a^3 \paren {3 s^2 + a^2} } {\paren {s^2 + a^2}^3} + \dfrac {5 a \paren {s^2 + a^2} } {\paren {s^2 - a^2}^2} }
      | c = simplifying
}}
{{eqn | r = \dfrac 1 {8 a} \paren {\dfrac {3 a \paren {s^2 - a^2}^2 + 2 a^3 \paren {3 s^2 + a^2} + 5 a \paren {s^2 - a^2} \paren {s^2 + a^2} } {\paren {s^2 - a^2}^3} }
      | c = common denominator
}}
{{eqn | r = \dfrac 1 {8 a} \paren {\dfrac {3 a s^4 - 6 a^3 s^2 + 3 a^5 + 6 a^3 s^2 + 2 a^5 + 5 a s^4 - 5 a^5} {\paren {s^2 - a^2}^3} }
      | c = multiplying out
}}
{{eqn | r = \dfrac 1 {8 a} \paren {\dfrac {8 a s^4} {\paren {s^2 - a^2}^3} }
      | c = simplifying
}}
{{eqn | r = \dfrac {s^4} {\paren {s^2 - a^2}^3}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Quantum Mechanics Commutator satisfies Jacobi Identity

Let $A, B, C$ be operators acting on the Hilbert Space of some quantum system.
Let $\sqbrk {\, \cdot, \cdot \,}$ denotes the commutator of these operators.
Then:
:$\sqbrk {A, \sqbrk {B, C} } + \sqbrk {B, \sqbrk {C, A} } + \sqbrk {C, \sqbrk {A, B} } = 0$
That is, the commutator satisfies the Jacobi identity.

Proof:
By expanding the definition of the commutator:
:$\sqbrk {A, B} = AB - BA$
into the definition of Jacobi identity, we obtain:
{{begin-eqn}}
{{eqn | l = \sqbrk {A, \sqbrk {B, C} } + \sqbrk {B, \sqbrk {C, A} } + \sqbrk {C, \sqbrk {A, B} }
      | r = \sqbrk {A, \paren {B C - C B} } + \sqbrk {B, \paren {C A - A C} } + \sqbrk {C, \paren {A B - B A} }
}}
{{eqn | r = \paren {A \paren {B C - C B} - \paren {B C - C B} A} + \paren {\paren {C A - A C} - \paren {C A - A C} B} + \paren {C \paren {A B - B A} - \paren {A B - B A} C}
}}
{{eqn | r = \paren {\paren {A B C - A C B} - \paren {B C A - C B A} } + \paren {\paren {B C A - B A C} - \paren {C A B - A C B} } + \paren {\paren {C A B - C B A} - \paren {A B C - B A C} }
}}
{{eqn | r = A B C - A C B - B C A + C B A + B C A - B A C - C A B + A C B + C A B - C B A - A B C + B A C
}}
{{eqn | r = 0
}}
{{end-eqn}}
Hence, the commutator satisfies the Jacobi identity.
{{qed}}
Theorem: Inverse Laplace Transform of s^5 over (s^2 - a^2)^3

:$\map {\laptrans {\dfrac {\paren {8 + a^2 t^2} \cosh a t + 7 a t \sinh a t} 8} } s = \dfrac {s^5} {\paren {s^2 - a^2}^3}$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {8 + a^2 t^2} \cosh a t + 7 a t \sinh a t} 8} } s
      | r = \dfrac 1 8 \paren {8 \map {\laptrans {\cosh a t} } s + a^2 \map {\laptrans {t^2 \cosh a t} } s + 7 a \map {\laptrans {t \sinh a t} } s}
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 8 \paren {8 \dfrac s {s^2 - a^2} + a^2 \map {\laptrans {t^2 \cosh a t} } s + 7 a \map {\laptrans {t \sinh a t} } s}
      | c = Laplace Transform of Hyperbolic Cosine
}}
{{eqn | r = \dfrac 1 8 \paren {\dfrac {8 s} {s^2 - a^2} + a^2 \map {\laptrans {t^2 \cosh a t} } s + 7 a \dfrac {2 a s} {\paren {s^2 - a^2}^2} }
      | c = Laplace Transform of $t \sinh a t$
}}
{{eqn | r = \dfrac 1 8 \paren {\dfrac {8 s} {s^2 - a^2} + a^2 \paren {\dfrac {2 s \paren {s^2 + 3 a^2} } {\paren {s^2 - a^2}^3} } + \dfrac {14 a^2 s} {\paren {s^2 - a^2}^2} }
      | c = Laplace Transform of $t^2 \cosh a t$
}}
{{eqn | r = \dfrac 1 8 \paren {\dfrac {8 s} {s^2 - a^2} + \dfrac {2 a^2 s \paren {s^2 + 3 a^2} } {\paren {s^2 - a^2}^3} + \dfrac {14 a^2 s} {\paren {s^2 - a^2}^2} }
      | c = simplifying
}}
{{eqn | r = \dfrac 1 8 \paren {\dfrac {8 s \paren {s^2 - a^2}^2 + 2 a^2 s \paren {s^2 + 3 a^2} + 14 a^2 s \paren {s^2 - a^2} } {\paren {s^2 - a^2}^3} }
      | c = common denominator
}}
{{eqn | r = \dfrac 1 8 \paren {\dfrac {8 s^5 - 16 a^2 s^3 + 8 a^4 s + 2 a^2 s^3 + 6 a^4 s + 14 a^2 s^3 - 14 a^4 s} {\paren {s^2 - a^2}^3} }
      | c = multiplying out
}}
{{eqn | r = \dfrac 1 8 \paren {\dfrac {8 s^5} {\paren {s^2 - a^2}^3} }
      | c = simplifying
}}
{{eqn | r = \dfrac {s^5} {\paren {s^2 - a^2}^3}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Laplace Transform of t^2 sinh a t over 2 a

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

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

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

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

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

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^2 \cosh a t} 2} } s
      | r = \dfrac 1 2 \paren {\map {\laptrans {t^2 \cosh a t} } s}
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 2 \paren {\dfrac {2 s \paren {s^2 + 3 a^2} } {\paren {s^2 - a^2}^3} }
      | c = Laplace Transform of $t^2 \cosh a t$
}}
{{eqn | r = \dfrac {s^3 + 3 a^2 s} {\paren {s^2 - a^2}^3}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Laplace Transform of t^3 cosh a t over 6

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

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^3 \cosh a t} 6} } s
      | r = \dfrac 1 6 \paren {\map {\laptrans {t^3 \cosh a t} } s}
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 6 \paren {\dfrac {6 \paren {s^4 + 6 a^2 s^2 + a^4} } {\paren {s^2 - a^2}^4} }
      | c = Laplace Transform of $t^3 \cosh a t$
}}
{{eqn | r = \dfrac {s^4 + 6 a^2 s^2 + a^4} {\paren {s^2 - a^2}^4}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}