Laplace transform is an effective method for solving ordinary and partial differential equations, and it has been successful in many applications. These equations describe how certain quantities change over time, such as the current in an electrical circuit, the vibrations of a membrane, or the flow of heat through a conductor. The Laplace transform helps converts differential equations into simpler algebraic equations. Both the Laplace transform and its inverse are important tools for analyzing dynamic control systems. The Laplace transform changes a signal in the time domain into a signal in the s-domain, also called the splane, which can then be solved by the formal rules of algebra.
Example
Coding
from sympy import *
t = Symbol('t', real=True)
s = Symbol('s', real=True, positive=True)
a = Symbol('a', real=True, positive=True)
f = t*exp(-a*t)
laplace_transform(f,t,s, noconds=True)
References:
1. https://www.geeksforgeeks.org/laplace-transform/
2. https://en.wikipedia.org/wiki/Laplace_transform
3. https://www.analog.com/media/en/technical-documentation/dsp-book/dsp_book_Ch32.pdf
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