avewright/proofwiki-math huggingface dataset:
Theorem: Laplace Transform of sine a t - a t cosine a t over 2 a^3

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

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

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

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

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

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

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

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

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

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {t \cos a t} } s
      | r = \map {\dfrac \d {\d s} } {\map {\laptrans {-\cos a t} } s}
      | c = Derivative of Laplace Transform
}}
{{eqn | r = -\map {\dfrac \d {\d s} } {\dfrac s {s^2 + a^2} }
      | c = Laplace Transform of Cosine
}}
{{eqn | r = -\dfrac {\paren {s^2 + a^2} \dfrac \d {\d s} s - s \map {\dfrac \d {\d s} } {s^2 + a^2} } {\paren {s^2 + a^2}^2}
      | c = Quotient Rule for Derivatives
}}
{{eqn | r = -\dfrac {s^2 + a^2 - 2 s^2} {\paren {s^2 + a^2}^2}
      | c = Power Rule for Derivatives and simplification
}}
{{eqn | r = \dfrac {s^2 - a^2} {\paren {s^2 + a^2}^2}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Laplace Transform of t sine a t

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

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {t \sin a t} } s
      | r = \map {\dfrac \d {\d s} } {\map {\laptrans {-\sin a t} } s}
      | c = Derivative of Laplace Transform
}}
{{eqn | r = -\map {\dfrac \d {\d s} } {\dfrac a {s^2 + a^2} }
      | c = Laplace Transform of Sine
}}
{{eqn | r = -\dfrac {\paren {s^2 + a^2} \dfrac \d {\d s} a - a \map {\dfrac \d {\d s} } {s^2 + a^2} } {\paren {s^2 + a^2}^2}
      | c = Quotient Rule for Derivatives
}}
{{eqn | r = -\dfrac {0 - 2 a s} {\paren {s^2 + a^2}^2}
      | c = Power Rule for Derivatives and simplification
}}
{{eqn | r = \dfrac {2 a s} {\paren {s^2 + a^2}^2}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Laplace Transform of t sine a t

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

Proof:
We have:
{{begin-eqn}}
{{eqn | l = \laptrans {\cos a t}
      | r = \int_0^\infty e^{-s t} \cos a t \rd t
      | c = {{Defof|Laplace Transform}}
}}
{{eqn | n = 1
      | r = \dfrac s {s^2 + a^2}
      | c = Laplace Transform of Cosine
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \dfrac \d {\d a} \int_0^\infty e^{-s t} \cos a t \rd t
      | r = \int_0^\infty e^{-s t} \paren {\map {\dfrac \partial {\partial a} } {\cos a t} } \rd t
      | c = Derivative of Integral
}}
{{eqn | r = \int_0^\infty e^{-s t} \paren {-t \sin a t} \rd t
      | c = Derivative of Cosine Function
}}
{{eqn | r = -\laptrans {t \sin a t}
      | c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \map {\dfrac \d {\d a} } {\dfrac s {s^2 + a^2} }
      | c = from $(1)$
}}
{{eqn | r = -\dfrac {2 a s} {\paren {s^2 + a^2}^2}
      | c = Quotient Rule for Derivatives
}}
{{end-eqn}}
and the result follows.
{{qed}}
Theorem: Laplace Transform of a t cosh a t - sinh a t over 2 a^3

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

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

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

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

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

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {t \sinh a t} } s
      | r = \map {\dfrac \d {\d s} } {\map {\laptrans {-\sinh a t} } s}
      | c = Derivative of Laplace Transform
}}
{{eqn | r = -\map {\dfrac \d {\d s} } {\dfrac a {s^2 - a^2} }
      | c = Laplace Transform of Hyperbolic Sine
}}
{{eqn | r = -\dfrac {\paren {s^2 - a^2} \dfrac \d {\d s} a - a \map {\dfrac \d {\d s} } {s^2 - a^2} } {\paren {s^2 - a^2}^2}
      | c = Quotient Rule for Derivatives
}}
{{eqn | r = -\dfrac {0 - 2 a s} {\paren {s^2 - a^2}^2}
      | c = Power Rule for Derivatives and simplification
}}
{{eqn | r = \dfrac {2 a s} {\paren {s^2 - a^2}^2}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Category:Laplace Transforms involving Hyperbolic Sine Function
Category:Inverse Laplace Transforms of Rational Functions
Category:Examples of Laplace Transforms
ccwqhg1008ncenf1xf3mck36klgtiy3
Theorem: Laplace Transform of sinh a t + a t cosh a t over 2 a

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

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

Let $\struct {R, +, \circ}$ be a ring.
Let $\struct {G, +_G}$ be an abelian group.
Let $\struct {G, +_G, \circ_G}_R$ be an $R$-module.
Let $\struct {H, +_H, \circ_H}_R$ be an $R$-submodule of $\struct {G, +_G, \circ_G}_R$.
Then $H$ is a subgroup of $G$.

Proof:
By definition of an $R$-submodule, $H$ must be a subset of $G$.
By definition of an $R$-submodule, $\struct {H, +_H, \circ_H}_R$ must be a $R$-module.
By definition of an $R$-module, $H$ must be a group.
Hence, by definition of a subgroup, $H$ is a subgroup of $G$.
{{qed}}
<!-- no sources because most authors require a subgroup in the definition of a submodule -->
Category:Submodules
Category:Subgroups
bmxxn8e70wu697ylyhritfubl3vuqll
Theorem: Laplace Transform of cosh a t + half a t sinh a t

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

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

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

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {t \cosh a t} } s
      | r = \map {\dfrac \d {\d s} } {\map {\laptrans {-\cosh a t} } s}
      | c = Derivative of Laplace Transform
}}
{{eqn | r = -\map {\dfrac \d {\d s} } {\dfrac s {s^2 - a^2} }
      | c = Laplace Transform of Hyperbolic Cosine
}}
{{eqn | r = -\dfrac {\paren {s^2 - a^2} \dfrac \d {\d s} s - s \map {\dfrac \d {\d s} } {s^2 - a^2} } {\paren {s^2 - a^2}^2}
      | c = Quotient Rule for Derivatives
}}
{{eqn | r = -\dfrac {s^2 - a^2 - 2 s^2} {\paren {s^2 - a^2}^2}
      | c = Power Rule for Derivatives and simplification
}}
{{eqn | r = \dfrac {s^2 + a^2} {\paren {s^2 - a^2}^2}
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Size of Maximal Planar Graph

Let $n$ be the order of a simple planar graph $G$ such that $n \ge 3$.
Let $\size G$ denote the size of $G$.
Then:
:$\size G = 3 n - 6$
{{iff}}:
:$G$ is maximal.
That is, a simple planar graph with $n$ vertices can have no more than $3 n - 6$ edges.

Proof:
{{ProofWanted}}
Category:Maximal Planar Graphs
3w0m06we1np9zjn4zjr6kac731ylv21
Theorem: Laplace Transform of t^2 sine a t

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

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {t^2 \sin a t} } s
      | r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t \sin 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 $t \sin a t$
}}
{{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 Sine Function
Category:Inverse Laplace Transforms of Rational Functions
Category:Examples of Laplace Transforms
ncao1uj9ftcl3532se06pgm2slmc5vh
Theorem: Inverse Laplace Transform of 1 over (s^2 + a^2)^3

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

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {3 - a^2 t^2} \sin a t - 3 a t \cos a t} {8 a^5} } } s
      | r = \dfrac 1 {8 a^5} \paren {3 \map {\laptrans {\sin a t} } s - a^2 \map {\laptrans {t^2 \sin a t} } s - 3 a \map {\laptrans {t \cos 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 \sin a t} } s - 3 a \map {\laptrans {t \cos a t} } s}
      | c = Laplace Transform of Sine
}}
{{eqn | r = \dfrac 1 {8 a^5} \paren {\dfrac {3 a} {s^2 + a^2} - a^2 \map {\laptrans {t^2 \sin a t} } s - \dfrac {3 a \paren {s^2 - a^2} } {\paren {s^2 + a^2}^2} }
      | c = Laplace Transform of $t \cos a t$
}}
{{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 $t^2 \sin a t$
}}
{{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: Laplace Transform of t^2 cosine a t

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

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {t^2 \cos a t} } s
      | r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t \cos 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 $t \cos a t$
}}
{{eqn | r = \map {\dfrac \d {\d s} } {\dfrac {a^2 - s^2} {\paren {s^2 + a^2}^2} }
      | c = simplifying
}}
{{eqn | r = \dfrac {\paren {s^2 + a^2}^2 \map {\dfrac \d {\d s} } {a^2 - s^2} - \paren {a^2 - s^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 {a^2 - s^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 {a^2 - s^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}}
Theorem: Inverse Laplace Transform of s over (s^2 + a^2)^3

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

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t \sin a t - a t^2 \cos a t} {8 a^3} } } s
      | r = \dfrac 1 {8 a^3} \paren {\map {\laptrans {t \sin a t} } s - a \map {\laptrans {t^2 \cos a t} } s}
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac {2 a s} {\paren {s^2 + a^2}^2} - a \map {\laptrans {t^2 \cos a t} } s}
      | c = Laplace Transform of $t \sin a t$
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac {2 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 $t^2 \cos a t$
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac {2 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^3} \paren {\dfrac {2 a s^3 + 2 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^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: Inverse Laplace Transform of s^2 over (s^2 + a^2)^3

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

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {1 + a^2 t^2} \sin a t - a t \cos a t} {8 a^3} } } s
      | r = \dfrac 1 {8 a^3} \paren {\map {\laptrans {\sin a t} } s + a^2 \map {\laptrans {t^2 \sin a t} } s - a \map {\laptrans {t \cos a t} } s}
      | c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac a {s^2 + a^2} + a^2 \map {\laptrans {t^2 \sin a t} } s - a \map {\laptrans {t \cos a t} } s}
      | c = Laplace Transform of Sine
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac a {s^2 + a^2} + a^2 \map {\laptrans {t^2 \sin a t} } s - \dfrac {a \paren {s^2 - a^2} } {\paren {s^2 + a^2}^2} }
      | c = Laplace Transform of $t \cos a t$
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac a {s^2 + a^2} + a^2 \paren {2 a \paren {\dfrac {3 s^2 - a^2} {\paren {s^2 + a^2}^3} } } - \dfrac {a \paren {s^2 - a^2} } {\paren {s^2 + a^2}^2} }
      | c = Laplace Transform of $t^2 \sin a t$
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac a {s^2 + a^2} + \dfrac {2 a^3 \paren {3 s^2 - a^2} } {\paren {s^2 + a^2}^3} - \dfrac {a \paren {s^2 - a^2} } {\paren {s^2 + a^2}^2} }
      | c = simplifying
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac {a \paren {s^2 + a^2}^2 + 2 a^3 \paren {3 s^2 - a^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 {a s^4 + 2 a^3 s^2 + a^5 + 6 a^3 s^2 - 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

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

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {3 t \sin a t + a t^2 \cos a t} {8 a} } } s
      | r = \dfrac 1 {8 a} \paren {3 \map {\laptrans {t \sin a t} } s + a \map {\laptrans {t^2 \cos 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 \cos a t} } s}
      | c = Laplace Transform of $t \sin a t$
}}
{{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 $t^2 \cos a t$
}}
{{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

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

Proof:
{{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {3 - a^2 t^2} \sin a t + 5 a t \cos a t} {8 a} } } s
      | r = \dfrac 1 {8 a} \paren {3 \map {\laptrans {\sin a t} } s - a^2 \map {\laptrans {t^2 \sin a t} } s + 5 a \map {\laptrans {t \cos 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 \sin a t} } s + 5 a \map {\laptrans {t \cos a t} } s}
      | c = Laplace Transform of Sine
}}
{{eqn | r = \dfrac 1 {8 a} \paren {\dfrac {3 a} {s^2 + a^2} - a^2 \map {\laptrans {t^2 \sin a t} } s + 5 \dfrac {a \paren {s^2 - a^2} } {\paren {s^2 + a^2}^2} }
      | c = Laplace Transform of $t \cos a t$
}}
{{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 $t^2 \sin a t$
}}
{{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}}