How to use the Laplace transform Calculator?
Here are step-by-step instructions on how to use the Laplace Transform Calculator:
Step-1: Enter the Function:
In the input box labeled “Enter the function f(t):”, type in the function f(t) for which you want to find the Laplace transform.
Step-2: Click “Calculate Laplace Transform”:
After entering the function, click on the “Calculate Laplace Transform” button.
How to find the Laplace transform?
To find the Laplace transform of a function, we can use the definition of the Laplace transform, which is:
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 the Laplace transform.
Find the Laplace transform of the function f(t)=t
Use the definition of the Laplace transform to find the Laplace transform of f(t)=t
F\left(s\right)=\int _0^{\infty }e^{-st}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)=\left[t\left(-\frac{1}{s}e^{-st}\right)-\int 1\left(-\frac{1}{s}e^{-st}\right)\right]_0^{\infty }\\
F\left(s\right)=\left[-\frac{te^{-st}}{s}-\int 1\left(-\frac{e^{-st}}{s}\right)\right]_0^{\infty }\\
F\left(s\right)=\left[-\frac{te^{-st}}{s}+\int \frac{e^{-st}}{s}\right]_0^{\infty }\\
F\left(s\right)=\left[-\frac{te^{-st}}{s}+\frac{1}{s}\int e^{-st}\right]_0^{\infty }\\
F\left(s\right)=\left[-\frac{te^{-st}}{s}+\frac{1}{s}\left(-\frac{1}{s}e^{-st}\right)\right]_0^{\infty }\\
F\left(s\right)=\left[-\frac{te^{-st}}{s}-\frac{1}{s^2}e^{-st}\right]_0^{\infty }\\
F\left(s\right)=-\frac{\infty e^{-s\cdot \infty }}{s}-\frac{1}{s^2}e^{-s\cdot \infty }-\left(-\frac{0e^{-s\cdot 0}}{s}-\frac{1}{s^2}e^{-s\cdot 0}\right)\\
F\left(s\right)=-\frac{0}{s}-\frac{1}{s^2}\cdot 0-\left(-\frac{0}{s}-\frac{1}{s^2}\cdot 1\right)\\
F\left(s\right)=0+0-\left(0-\frac{1}{s^2}\right)\\
F\left(s\right)=-\left(-\frac{1}{s^2}\right)\\
F\left(s\right)=\frac{1}{s^2}\\
FAQs on the Laplace transform:
1) Who invented the Laplace transform?
Answer: The concept of the Laplace transform is named after Pierre-Simon Laplace, a French mathematician and astronomer who lived from 1749 to 1827
2) What is the Laplace Transform?
Answer: The Laplace transform is an integral transform used to convert a function of time f(t) into a function of a complex variable s. It is widely used in engineering and mathematics for solving differential equations, analyzing systems, and solving initial value problems.
3) Why is the Laplace Transform Useful?
Answer: The Laplace transform is useful because it simplifies the process of solving differential equations. It transforms differential equations into algebraic equations, which are often easier to solve. Additionally, it provides a convenient way to analyze systems in the frequency domain.
How is the Laplace Transform denoted?
4) How is the Laplace Transform denoted?
Answer: The Laplace transform of a function f(t) is denoted by F(s), where s is a complex variable. Mathematically, it is represented as: F\left(s\right)=\int _0^{\infty }e^{-st}f\left(t\right)dt
5) What is the Laplace of 1?
Answer: The Laplace transform of the constant function 1 is: L\left\{1\right\}=\frac{1}{s}
