Laplace Transform

Laplace transform is named in honour of the great French mathematician, Pierre Simon De Laplace (1749-1827). Like all transforms, the Laplace transform changes one signal into another according to some fixed set of rules or equations. The best way to convert differential equations into algebraic equations is the use of Laplace transformation. In this section, students get a step-by-step explanation for every concept and will find it extremely easy to understand this topic in a detailed way.

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

Laplace transform

whenever the improper integral converges.

Properties of Laplace Transform

Some of the Laplace transformation properties are:

If f1 (t) ⟷ F1 (s) and [note: ⟷ implies Laplace Transform]

f2 (t) ⟷ F2 (s), then

Linearity Property A f1(t) + B f2(t) ⟷ A F1(s) + B F2(s)
Frequency Shifting Property es0t f(t)) ⟷ F(s – s0)
nth Derivative Property (dn f(t)/ dtn) ⟷ sn F(s) − ni = 1 sn − i fi − 1 (0)
Integration t0 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 tn at t = 1,2,3,… n!/s(n+1) 12 tp, 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)/(2n s(n+1/2)
4 sin(at) a/(s2+a2) 14 cos(at) s/(s2+a2)
5 t sin(at) 2as/(s2+a2)2 15 t cos(at) (s2-a2)/(s2+a2)2
6 sin(at+b) (s sin(b)+ a cos(b)/(s2+a2) 16 cos(at+b) (s cos(b)-a sin(b)/(s2+a2)
7 sinh(at) a/(s2-a2) 17 cosh(at) s/(s2-a2)
8 e(at)sin(bt) b/((s-a)2+b2) 18 e(at)cos(bt) (s-a)/((s-a)2+b2)
9 e(ct)f(t) F(s-c) 19 tnf(t) at n = 1,2,3.. (-1)n Fn s
10 f'(t) sF(s) – f(0) 20 f”(t) s2F(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 = e3x 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(e3x)

L(y’) – L(2y) = 1/(s-3)

(because L(eax) = 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) = -6e2x + e3x

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 {(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. To learn more in detail visit the link given for inverse laplace transform.

Laplace Transform Examples

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

Laplace transform Example

Laplace Transform

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