Laplace Transform/Examples/Example 1
Jump to navigation
Jump to search
Example of Laplace Transform
Let $\laptrans f$ denote the Laplace transform of the real function $f$.
- $\laptrans {4 e^{5 t} + 6 t^3 - 3 \sin 4 t + 2 \cos 2 t} = \dfrac 4 {s - 5} + \dfrac {36} {s^4} - \dfrac {12} {s^2 + 16} + \dfrac {2 s} {s^2 + 4}$
Proof
\(\ds \laptrans {4 e^{5 t} + 6 t^3 - 3 \sin 4 t + 2 \cos 2 t}\) | \(=\) | \(\ds 4 \laptrans {e^{5 t} } + 6 \laptrans {t^3} - 3 \laptrans {\sin 4 t} + 2 \laptrans {\cos 2 t}\) | Linear Combination of Laplace Transforms | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \paren {\dfrac 1 {s - 5} } + 6 \laptrans {t^3} - 3 \laptrans {\sin 4 t} + 2 \laptrans {\cos 2 t}\) | Laplace Transform of Exponential | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \paren {\dfrac 1 {s - 5} } + 6 \paren {\dfrac {3!} {s^4} } - 3 \laptrans {\sin 4 t} + 2 \laptrans {\cos 2 t}\) | Laplace Transform of Positive Integer Power | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \paren {\dfrac 1 {s - 5} } + 6 \paren {\dfrac {3!} {s^4} } - 3 \paren {\dfrac 4 {s^2 + 4^2} } + 2 \laptrans {\cos 2 t}\) | Laplace Transform of Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \paren {\dfrac 1 {s - 5} } + 6 \paren {\dfrac {3!} {s^4} } - 3 \paren {\dfrac 4 {s^2 + 4^2} } + 2 \paren {\dfrac s {s^2 + 2^2} }\) | Laplace Transform of Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 4 {s - 5} + \dfrac {36} {s^4} - \dfrac {12} {s^2 + 16} + \dfrac {2 s} {s^2 + 4}\) | simplifying |
$\blacksquare$
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: The Linearity Property: $6$