Laplace transform
The Laplace transform of a function f(t) is calculated as
\[ F(s) = \int_{0}^{\infty} f(t)e^{-st}\,dt \]
Notation:
\[ F(s) = \mathcal{L}\{f(t)\} \]
Inverse transform:
\[ f(t) = \mathcal{L}^{-1}\{F(s)\} \]
Example: Find the Laplace transform of f(t) = a
View the solutionExample: Find the Laplace transform of f(t) = 2 + 5t
View the solutionExample: Find the Laplace transform of f(t) = t
View the solutionInverse Laplace Transform Problem
Find the inverse Laplace transform of F(s) = (3s - 4) / (s2 + 9)
View the solutionSolving linear ODEs using the Laplace transform
Problem 1: Solve the ODE 2 0 with the initial condition df/dt + 2f = 0 with the initial condition f(0) = 5
View the solutionProblem 2: Solve the ODE y" + 4y = 0 with the initial conditions y(0) = 0 y'(0) = 0
View the solutionProblem 3: Solve the ODE f" + 4f = 4cos(3t) using the initial conditions f(0) = 0 and f'(0) = 1
View the solutionLaplace transform: Test yourself
Using the table below:
Test your knowledge as follows
View the exercisesView the solutions part 1
View the solutions part 2
View the solutions part 3
View the solutions part 4
View the solutions part 5
Useful Video tutorials on Partial Fraction Decomposition:
▶ Watch Video Tutorial ▶ Watch Video Tutorial ▶ Watch Video TutorialFrequently Asked Questions
Q: What is the Laplace Transform?
A: The Laplace Transform converts a function of time, f(t), into a function of a complex variable s, written as F(s).
It is defined by the integral \( F(s) = \int_0^\infty f(t) e^{-st} \, dt \). It is widely used in engineering and mathematics to simplify the process of solving differential equations.
Q: Why is the Laplace Transform useful?
A: The Laplace Transform converts differential equations into algebraic equations, which are much easier to solve.
After solving in the s-domain, the Inverse Laplace Transform converts the result back into a function of time.
Q: What is the Inverse Laplace Transform?
A: The Inverse Laplace Transform converts a function in the s-domain, F(s), back into a function of time, f(t).
It is written as f(t) = ℒ⁻¹{F(s)} and is typically evaluated using a table of standard Laplace Transform pairs.
Q: How do you use the Laplace Transform to solve a linear ODE?
A: Apply the Laplace Transform to both sides of the equation to convert it into an algebraic equation in s.
Substitute the initial conditions, solve for F(s), then apply the Inverse Laplace Transform to find f(t).
Q: What is partial fraction decomposition and why is it used with Laplace Transforms?
A: Partial fraction decomposition breaks a complex rational expression in s into simpler fractions that match standard Laplace Transform pairs from a table.
This makes it possible to apply the Inverse Laplace Transform to each term individually.