Laplace Transform Calculator
Complete Laplace transform reference with 18 common transform pairs, 10 key properties, and 5 worked inverse transform examples. Searchable, filterable, and ready for your differential equations homework.
How to Use the Laplace Transform Calculator
Browse the Transform Table
The main tab shows 18 common Laplace transform pairs organized by category: Basic, Exponential, Trigonometric, Hyperbolic, and Special functions. Use the search bar or category filters to find the transform you need.
Review Key Properties
Switch to the Properties tab to see 10 essential properties including linearity, derivative transforms, shifting theorems, convolution, and scaling. Each property includes the formula and a plain-English explanation.
Study Inverse Examples
The Inverse Examples tab walks through 5 worked problems showing how to go from F(s) back to f(t) using partial fractions, direct lookup, and the shifting theorem. Each example shows the method used.
Complete Guide to Laplace Transforms
The Big Idea Behind the Laplace Transform
Imagine you have a differential equation describing how current flows through a circuit after you flip a switch. Solving it directly means finding a function whose derivatives satisfy the equation, which often involves guessing a solution form and grinding through algebra. The Laplace transform offers a shortcut: convert everything to the s-domain, where derivatives become multiplications, solve the resulting polynomial, and convert back.
The mathematical definition is straightforward. You take your time-domain function f(t), multiply it by the decaying exponential e to the negative st, and integrate from zero to infinity. The result, F(s), is a new function that encodes the same information as f(t) but in a form where calculus operations become algebra. Pierre-Simon Laplace introduced this idea in the late 1700s, and it has been a cornerstone of applied mathematics ever since.
How to Solve a Differential Equation with Laplace Transforms
Here is the typical workflow. Start with a differential equation like y double prime plus 3y prime plus 2y equals e to the negative t, with initial conditions y(0) = 1 and y prime of 0 equals 0. Take the Laplace transform of every term. The derivative property converts y double prime into s squared Y(s) minus s times 1 minus 0, and y prime into s Y(s) minus 1. The right side transforms to 1 over (s plus 1).
Now you have an algebraic equation in Y(s). Solve for Y(s), which gives a ratio of polynomials. Use partial fraction decomposition to break it into pieces like A over (s plus 1) plus B over (s plus 1) squared plus C over (s plus 2). Look up each piece in the transform table to get the time-domain answer. The entire process is mechanical once you know the steps, which is why it is taught in every engineering program.
Transfer Functions and Control Theory
In control engineering, the Laplace transform is not just a solution technique; it is the language of the field. A transfer function H(s) describes how a system transforms an input signal into an output signal, entirely in the s-domain. The poles of H(s), meaning the values of s where the denominator is zero, determine whether the system is stable. If all poles have negative real parts, the system is stable. If any pole has a positive real part, the output grows without bound.
This framework lets engineers design controllers by manipulating transfer functions algebraically. PID controllers, lead-lag compensators, and state-space models all rely on the Laplace transform. Even modern digital control systems use the closely related Z-transform, which is essentially the Laplace transform adapted for discrete-time signals.
Laplace Transform Calculator FAQ
Common questions about Laplace transforms, inverse transforms, and differential equations.
Disclaimer: This Laplace Transform Calculator provides a reference table and worked examples. For symbolic computation of arbitrary transforms, use a computer algebra system like Wolfram Alpha or MATLAB.