Exercise · ZW-LAPLACE-TRANSFORM-006 · ★★☆ Intermediate
Compute \(\mathcal L\{t^2\}(s)\).
Choose one option.
Show answer
- A. \(\tfrac2{s^3}\)
Explanation
\(\mathcal L\{t^n\}=n!/s^{n+1}\); with \(n=2\) we get \(2/s^3\). The value \(6/s^4\) would be for \(t^3\).
Review the concept: Laplace transform
The Laplace transform \(\mathcal L\) assigns to a function \(f(t)\), defined for \(t\ge0\), the function \(F(s)\) obtained by integrating \(f\) times \(e^{-st}\) over \([0,\infty)\).
\[\mathcal L\{f\}(s)=F(s)=\int_0^\infty e^{-st}f(t)\,dt\]
More practice problems: laplace transform
- ZW-LAPLACE-TRANSFORM-001 Compute mathcal L\1\(s) for s>0.
- ZW-LAPLACE-TRANSFORM-002 Compute mathcal L\t\(s).
- ZW-LAPLACE-TRANSFORM-003 Compute mathcal L\e^3t\(s) for s>3.
- ZW-LAPLACE-TRANSFORM-004 Compute mathcal L\sin 3t\(s).
- ZW-LAPLACE-TRANSFORM-005 Compute mathcal L\cos 2t\(s).
- ZW-LAPLACE-TRANSFORM-007 Use linearity to compute mathcal L\3+2t\(s).
- ZW-LAPLACE-TRANSFORM-008 Compute mathcal L\te^t\(s).
- ZW-LAPLACE-TRANSFORM-009 For which values of s does the integral defining mathcal L\e^2t\(s) converge?