The **Laplace transform **is a technique for solving differential equations. Here differential equation of time domain form is first transformed to algebraic equation of frequency domain form. After solving the algebraic equation in frequency domain, the result then is finally transformed to time domain form to achieve the ultimate solution of the differential equation

## History

Transformation in mathematics deals with the conversion of one function to another function that may not be in the same domain. The transform method finds its application in those problems which can’t be solved directly. This transform is named after the mathematician and renowned astronomer Pierre Simon Laplace who lived in France.

He used a similar transform on his additions to the probability theory. It became popular after World War Two. This transform was made popular by Oliver Heaviside, an English Electrical Engineer. Other famous scientists such as Niels Abel, Mathias Lerch, and Thomas Bromwich used it in the 19th century.

The complete history of the Laplace Transforms can be tracked a little more to the past, more specifically 1744. This is when another great mathematician called Leonhard Euler was researching on other types of integrals. Euler however did not pursue it very far and left it. An admirer of Euler called Joseph Lagrange; made some modifications to Euler’s work and did further work. LaGrange’s work got Laplace’s attention 38 years later, in 1782 where he continued to pick up where Euler left off. But it was not 3 years later; in 1785 where Laplace had a stroke of genius and changed the way we solve differential equations forever. He continued to work on it and continued to unlock the true power of the Laplace transform until 1809, where he started to use infinity as a integral condition.

## Laplace Transform Formula

Laplace transform is the integral transform of the given derivative function with real variable t to convert into complex function with variable s. For t ≥ 0, let f(t) be given and assume the function satisfies certain conditions to be stated later on.

The Laplace transform of f(t), that it is denoted by **f(t) or F(s)** is defined by the equation

whenever the improper integral converges.

**Standard notation:** Where the notation is clear, we will use an upper case letter to indicate the Laplace transform, e.g, L(f; s) = F(s).

The Laplace transform we defined is sometimes called the **one-sided Laplace transform**. There is a two-sided version where the integral goes from −∞ to ∞.

## Properties of Laplace Transform

Some of the Laplace transformation properties are:

If f_{1} (t) ⟷ F_{1} (s) and [note: ⟷ implies Laplace Transform]

f_{2} (t) ⟷ F_{2} (s), then

Linearity Property | A f_{1}(t) + B f_{2}(t) ⟷ A F_{1}(s) + B F_{2}(s) |

Frequency Shifting Property | e^{s}_{0}t f(t)) ⟷ F(s – s_{0}) |

nth Derivative Property | (d^{n} f(t)/ dt^{n}) ⟷ s^{n} F(s) − ^{n}∑_{i = 1 }s^{n − i} f^{i − 1} (0^{−}) |

Integration | ^{t}∫_{0} f(λ) dλ ⟷ 1⁄s F(s) |

Multiplication by Time | T f(t) ⟷ (−d F(s)⁄ds) |

Complex Shift Property | f(t) e^{−at} ⟷ F(s + a) |

Time Reversal Property | f (-t) ⟷ F(-s) |

Time Scaling Property | f (t⁄a) ⟷ a F(as) |

## Laplace Transform Table

Sl No. | f(t) | L(f(t)) = F(s) | Sl No. | f(t) | L(f(t)) = F(s) |

1 | 1 | 1/s | 11 | e^{(at)} | 1/(s − a) |

2 | t^{n} at t = 1,2,3,… | n!/s^{(n+1)} | 12 | t^{p}, at p>-1 | Γ(p+1)/s^{(p+1)} |

3 | √(t) | √π/2s^{(3/2)} | 13 | t^{(n-1/2)} at n = 1,2,.. | (1.3.5…(2n-1)√π)/(2^{n} s^{(n+1/2)} |

4 | sin(at) | a/(s^{2}+a^{2}) | 14 | cos(at) | s/(s^{2}+a^{2}) |

5 | t sin(at) | 2as/(s^{2}+a^{2})^{2} | 15 | t cos(at) | (s^{2}-a^{2})/(s^{2}+a^{2})^{2} |

6 | sin(at+b) | (s sin(b)+ a cos(b)/(s^{2}+a^{2}) | 16 | cos(at+b) | (s cos(b)-a sin(b)/(s^{2}+a^{2}) |

7 | sinh(at) | a/(s^{2}-a^{2}) | 17 | cosh(at) | s/(s^{2}-a^{2}) |

8 | e^{(at)}sin(bt) | b/((s-a)^{2}+b^{2}) | 18 | e^{(at)}cos(bt) | (s-a)/((s-a)^{2}+b^{2}) |

9 | e^{(ct)}f(t) | F(s-c) | 19 | t^{n}f(t) at n = 1,2,3.. | (-1)^{n} F^{n} s |

10 | f'(t) | sF(s) – f(0) | 20 | f”(t) | s^{2}F(s) − sf(0) − f'(0) |

## Laplace Transform of Differential Equation

The Laplace transform is a well established mathematical technique for solving a differential equation. Many mathematical problems are solved using transformations. The idea is to transform the problem into another problem that is easier to solve. On the other side, the inverse transform is helpful to calculate the solution to the given problem.

For better understanding, let us solve a first-order differential equation with the help of Laplace transformation,

Consider y’- 2y = e^{3x} and y(0) = -5. Find the value of L(y).

First step of the equation can be solved with the help of the linearity equation:

L(y’ – 2y] = L(e^{3x})

L(y’) – L(2y) = 1/(s-3)

(because L(e^{ax}) = 1/(s-a))

L(y’) – 2s(y) = 1/(s-3)

sL(y) – y(0) – 2L(y) = 1/(s-3)

(Using Linearity property of the Laplace transform)

L(y)(s-2) + 5 = 1/(s-3) (Use value of y(0) ie -5 (given))

L(y)(s-2) = 1/(s-3) – 5

L(y) = (-5s+16)/(s-2)(s-3) …..(1)

here (-5s+16)/(s-2)(s-3) can be written as -6/s-2 + 1/(s-3) using partial fraction method

(1) implies L(y) = -6/(s-2) + 1/(s-3)

L(y) = -6e^{2x} + e^{3x}

## Laplace Transform Examples

Below examples are based on some important elementary functions of Laplace transform.

## Inverse Laplace Transform

The inverse of complex function F(s) to produce a real valued function f(t) is inverse laplace transformation of the function. If a unique function is continuous on o to ∞ limit and have the property of Laplace Transform,

*F***( s) = L {f (t)} (s);**

is said to be Inverse laplace transform of F(s). It can be written as, L^{-1} [f(s)] (t). This function is exponentially restricted real function.

## Applications

- It is used to convert complex differential equations to a simpler form having polynomials.
- It is used on to convert derivatives into multiple of domain variable and then convert the polynomials back to the differential equation using Inverse Laplace transform.
- It is used in the telecommunication field to send signals both the sides of the medium. For example, when the signals are sent through phone then they are first converted into a time-varying wave and then super-imposed on the medium.
- It is also used for many engineering tasks such as Electrical Circuit Analysis, Digital Signal Processing, System Modelling, etc.

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