How to use Laplace transform Calculator for Piecewise functions?
Here are step-by-step instructions on how to use the Laplace transform Calculator for Piecewise functions:
Step-1: Enter Function 1:
In the input box labeled “Function 1”, type the first function of t.
Step-2: Enter interval of Function 1:
In the input box labeled “Interval”, type the interval. For example, 0<=t<1.
Step-3: Enter Function 2:
In the input box labeled “Function 2”, type the first function of t.
Step-4: Enter interval of Function 2:
In the input box labeled “Interval”, type the interval. For example, t>=1.
How to find the Laplace transform of Piecewise functions?
To find the Laplace transform of Piecewise functions, we can use the definition of the Laplace transform, which is given by:
F\left(s\right)=\int _0^{\infty }e^{-st}f\left(t\right)dt
where F(s) is the Laplace transform of the function f(t), and s is a complex variable.
Example on how to find the Laplace transform of Piecewise functions.
Consider the Piecewise function below.
f\left(t\right)=\left\{\begin{matrix}1,\:\:\:\:\:0\le t<1\\ t,\:\:\:\:\:\:\:\:\:\:\:\:t\ge 1\end{matrix}\right.
Question: For the Piecewise function above, find its Laplace transform
Use the definition of the Laplace transform to find the Laplace transform of the Piecewise function f(t)=t.
F\left(s\right)=\int _0^{\infty }e^{-st}f(t)dt\\
F\left(s\right)=\int _0^1e^{-st}\cdot 1dt+\int _1^{\infty }e^{-st}\cdot tdt \\
F\left(s\right)=\int _0^1e^{-st}dt+\int _1^{\infty }e^{-st}\cdot tdt \\
F\left(s\right)=\left[-\frac{e^{-st}}{s}\right]_0^1+\int _1^{\infty }e^{-st}\cdot tdt \\
F\left(s\right)=-\frac{1}{s}\left[e^{-st}\right]_0^1+\int _1^{\infty }e^{-st}\cdot tdt \\
F\left(s\right)=-\frac{1}{s}\left[e^{-s\cdot 1}-e^{-s\cdot 0}\right]+\int _1^{\infty }e^{-st}\cdot tdt \\
F\left(s\right)=-\frac{1}{s}\left[e^{-s}-e^0\right]+\int _1^{\infty }e^{-st}\cdot tdt \\
F\left(s\right)=-\frac{1}{s}\left(e^{-s}-1\right)+\int _1^{\infty }e^{-st}\cdot tdt \\
Use the integration by parts method.
Substitute
u=t\:\:\:\:\:\:\:and\:\:\:\:\:\:\:\:\:v'=e^{-st}\\
u'=\frac{d}{dt}\left(t\right)\:\:\:\:\:\:\:and\:\:\:\:\:\:\:\:\:v=\int e^{-st}\\
u'=1\:\:\:\:\:\:\:and\:\:\:\:\:\:\:\:\:v=-\frac{1}{s}e^{-st}\\
The rule for integration by parts is given by, \int uv'=uv-\int u'v
F\left(s\right)=-\frac{1}{s}\left(e^{-s}-1\right)+\left[t\left(-\frac{1}{s}e^{-st}\right)-\int 1\left(-\frac{1}{s}e^{-st}\right)\right]_1^{\infty }\\
F\left(s\right)=-\frac{1}{s}\left(e^{-s}-1\right)+\left[-\frac{te^{-st}}{s}+\frac{1}{s}\int e^{-st}\right]_1^{\infty }\\
F\left(s\right)=-\frac{1}{s}\left(e^{-s}-1\right)+\left[-\frac{te^{-st}}{s}+\frac{1}{s}\left(-\frac{e^{-st}}{s}\right)\right]_1^{\infty }\\
F\left(s\right)=-\frac{1}{s}\left(e^{-s}-1\right)+\left[-\frac{te^{-st}}{s}-\frac{e^{-st}}{s^2}\right]_1^{\infty }\\
F\left(s\right)=-\frac{1}{s}\left(e^{-s}-1\right)+\left[-\frac{\infty e^{-s\cdot \infty }}{s}-\frac{1}{s^2}e^{-s\cdot \infty }-\left(-\frac{1e^{-s\cdot 1}}{s}-\frac{1}{s^2}e^{-s\cdot 1}\right)\right]\\
F\left(s\right)=-\frac{1}{s}\left(e^{-s}-1\right)+\left[-\frac{\infty e^{-s\cdot \infty }}{s}-\frac{1}{s^2}e^{-s\cdot \infty }-\left(-\frac{1e^{-s\cdot 1}}{s}-\frac{1}{s^2}e^{-s\cdot 1}\right)\right]\\
F\left(s\right)=-\frac{1}{s}\left(e^{-s}-1\right)+\left[-\frac{0}{s}-\frac{1}{s^2}\cdot 0-\left(-\frac{e^{-s}}{s}-\frac{1}{s^2}e^{-s}\right)\right]\\
F\left(s\right)=-\frac{1}{s}\left(e^{-s}-1\right)+\left[-0-0-\left(-\frac{e^{-s}}{s}-\frac{e^{-s}}{s^2}\right)\right]\\
F\left(s\right)=-\frac{1}{s}\left(e^{-s}-1\right)+\frac{e^{-s}}{s}+\frac{e^{-s}}{s^2}\\
F\left(s\right)=-\frac{e^{-s}}{s}+\frac{1}{s}+\frac{e^{-s}}{s}+\frac{e^{-s}}{s^2}\\
F\left(s\right)=\frac{1}{s}+\frac{e^{-s}}{s^2}\\
F\left(s\right)=\frac{e^{-s}}{s^2}+\frac{1}{s}\\
