Laplace Transform of t^2 by Cosine a t

Theorem

Let $\sin$ denote the real sine function.

Let $\laptrans f$ denote the Laplace transform of a real function $f$.

Then:

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

Proof

\(\ds \laptrans {t^2 \cos a t}\)

\(=\)

\(\ds -\map {\dfrac {\d^2} {\d s^2} } {\laptrans {\cos a t} }\)

Higher Order Derivatives of Laplace Transform

\(\ds \)

\(=\)

\(\ds -\map {\dfrac {\d^2} {\d s^2} } {\dfrac a {s^2 + a^2} }\)

Laplace Transform of Sine

\(\ds \)

\(=\)

\(\ds \dfrac {2 s^3 - 6 a^2 s} {\paren {s^2 + a^2}^3}\)

Quotient Rule for Derivatives

$\blacksquare$

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Multiplication by Powers of $t$: $20 \ \text{(b)}$
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of Special Laplace Transforms: $32.59$