Laplace Transform of Exponential times Function/Examples/Example 3

Examples of Use of Laplace Transform of Exponential times Function

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

$\laptrans {e^{-2 t} \sin 4 t} = \dfrac 4 {s^2 + 4 s + 20}$

Proof

\(\ds \laptrans {\sin 4 t}\)

\(=\)

\(\ds \dfrac 4 {s^2 + 4^2}\)

Laplace Transform of Sine

\(\ds \leadsto \ \ \)

\(\ds \laptrans {e^{-2 t} \sin 4 t }\)

\(=\)

\(\ds \dfrac 4 {\paren {s + 2}^2 + 16}\)

Laplace Transform of Exponential times Function

\(\ds \)

\(=\)

\(\ds \dfrac 4 {s^2 + 4 s + 20}\)

multiplying out

$\blacksquare$

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Translation and Change of Scale Properties: $8 \ \text{(b)}$