Function f(t)	Laplace Transform F(s)
1	[Tex]\frac{1}{s}[/Tex]
t	[Tex]\frac{1}{s^2}[/Tex]
[Tex]t^n[/Tex]	[Tex]\frac{n!}{s^{n+1}}[/Tex]
[Tex]e^{at}[/Tex]	[Tex]\frac{1}{s-a}[/Tex]
[Tex]t e^{at}[/Tex]	[Tex]\frac{1}{(s-a)^2}[/Tex]
[Tex]t^n e^{at}[/Tex]	[Tex]\frac{n!}{(s-a)^{n+1}}[/Tex]
[Tex]sin(at)[/Tex]	[Tex]\frac{a}{s^2 + a^2}[/Tex]
[Tex]cos(at)[/Tex]	[Tex]\frac{s}{s^2 + a^2}[/Tex]
[Tex]sinh(at)[/Tex]	[Tex]\frac{a}{s^2 – a^2}[/Tex]
[Tex]\cos h(at)[/Tex]	[Tex]\frac{s}{s^2 – a^2}[/Tex]
[Tex]e^{at} \sin(bt)[/Tex]	[Tex]\frac{b}{(s-a)^2 + b^2}[/Tex]
[Tex]e^{at} \cos(bt)[/Tex]	[Tex]\frac{s-a}{(s-a)^2 + b^2}[/Tex]
[Tex]u(t-a)[/Tex]	[Tex]\frac{e^{-as}}{s}[/Tex]
[Tex]\delta(t-a)[/Tex]	[Tex]e^{-as}[/Tex]
[Tex]\int_0^t f(\tau) d\tau[/Tex]	[Tex]\frac{F(s)}{s}[/Tex]
[Tex]f'(t)[/Tex]	[Tex]sF(s) – f(0)[/Tex]
[Tex]f”(t)[/Tex]	[Tex]s^2F(s) – sf(0) – f'(0)[/Tex]

Laplace Transforms example Problems: Solved

Example 1: Find the Laplace transform of f(t) = 3t².

Solution:

F(s) = 6/s³

Example 2: Calculate the Laplace transform of f(t) = e^(-2t).

Solution:

F(s) = 1/(s+2)

Example 3: What is the Laplace transform of f(t) = sin(3t)?

Solution:

F(s) = 3/(s² + 9)

Example 4: Determine the Laplace transform of f(t) = 4 cos(5t).

Solution:

F(s) = 4s/(s² + 25)

Example 5: Find the Laplace transform of f(t) = t³.

Solution:

F(s) = 6/s⁴

Example 6: Calculate the Laplace transform of f(t) = 2e^(3t).

Solution:

F(s) = 2/(s-3)

Laplace Transforms Books

• The Laplace Transform: Theory and Applications by Joel L. Schiff
• Laplace Transforms and Their Applications to Differential Equations by N.W. McLachlan
• Operational Mathematics by Ruel V. Churchill
• A First Course in Laplace Transforms by J. C. Jaeger
• Applied Laplace Transforms and z-Transforms for Scientists and Engineers by Urs Graf

Laplace Transforms Practice Problems : Unsolved

Q1. Find the Laplace transform of f(t) = 3e^(2t) – 5sin(4t).

Q.2 Determine the inverse Laplace transform of F(s) = (s² + 4) / (s² + 2s + 5).

Q.3 Solve the initial value problem:

y” + 4y’ + 4y = e^(-2t) Where y(0) = 1 and y'(0) = 0.

Q.4 Find the Laplace transform of the periodic function f(t) = t for 0 ≤ t < 2, and f(t + 2) = f(t) for all t ≥ 0.

Q.5 Use the Laplace transform to solve the integro-differential equation:

y'(t) + ∫(0 to t) y(τ)dτ = 1 where y(0) = 0.

Q.6 Find the Laplace transform of the unit step function u(t-3) multiplied by e^(-2t).

Q.7 Determine the inverse Laplace transform of F(s) = ln((s² + 1)/(s² + 4)).

Q.8 Solve the system of differential equations using Laplace transforms:

x'(t) = 2x(t) – y(t) and y'(t) = x(t) + 2y(t) with initial conditions x(0) = 1, y(0) = 0.

Laplace Transforms Practice Problems – FAQs

What is the Laplace transform?

The Laplace transform is an integral transform used to convert a function of time into a function of complex frequency. It’s widely used in mathematics, physics, and engineering to simplify the solution of differential equations.

Who invented the Laplace transform?

The Laplace transform is named after Pierre-Simon Laplace, a French mathematician who introduced this transform in his work on probability theory in the late 18th century.

What are the main applications of Laplace transforms?

Laplace transforms are commonly used in solving differential equations, analyzing electrical circuits, control systems, signal processing, and various areas of applied mathematics and engineering.

What is the notation for the Laplace transform?

The Laplace transform of a function f(t) is typically denoted as F(s) or L{f(t)}, where s is the complex frequency variable.

What is the inverse Laplace transform?

The inverse Laplace transform is an operation that converts a function in the complex frequency domain back to the time domain. It’s denoted as L^(-1){F(s)} or f(t) = L^(-1){F(s)}.

What is the Laplace transform of a constant?

The Laplace transform of a constant k is k/s, where s is the complex frequency variable.

What is the convolution theorem in Laplace transforms?

The convolution theorem states that the Laplace transform of the convolution of two functions is equal to the product of their individual Laplace transforms.