Laplace Transform Table and Examples

The Laplace transform is one of the most powerful tools in applied mathematics. It converts a function of time into a function of a complex variable, turning differential equations into algebraic equations that are far easier to solve. Engineers, physicists, and mathematicians rely on it daily to analyse circuits, control systems, mechanical vibrations, and signal processing problems.

This guide covers the definition, the most important transform pairs, key properties, inverse transforms, and two fully worked examples of solving ordinary differential equations (ODEs) using the Laplace method.

Definition of the Laplace Transform

Let be a function defined for . Its Laplace transform is the function defined by:

The integral converges for values of in some right half-plane , where depends on the growth rate of . In practice, you rarely evaluate the improper integral by hand. Instead, you use a table of known transform pairs and a set of properties that let you build up transforms of more complicated functions from simpler ones.

Table of Common Laplace Transforms

The following pairs form the backbone of Laplace transform work. Memorise these and you can handle the vast majority of textbook problems.

Key Properties

Linearity

The Laplace transform is linear, meaning:

This lets you transform each term of a sum separately and combine the results.

First Shifting Theorem (s-shifting)

If , then:

Multiplying by an exponential in the time domain shifts the transform in the -domain. This is how we derive the transforms of and from the basic sine and cosine transforms.

Transform of Derivatives

This is the property that makes solving ODEs possible:

Differentiation in the time domain becomes multiplication by in the frequency domain, minus initial conditions. This is precisely why the Laplace transform converts differential equations into algebraic equations: derivatives vanish, replaced by powers of and known initial values.

Transform of Integrals

Integration in the time domain becomes division by .

Inverse Laplace Transform

The inverse Laplace transform recovers from . Formally it involves a contour integral in the complex plane, but in practice you use partial fraction decomposition to break into pieces that match the standard table entries above.

Example. Find the inverse Laplace transform of:

Reading from the table: corresponds to , and corresponds to . By linearity:

Solving ODEs with Laplace Transforms: The Method

The procedure has four steps:

1. Take the Laplace transform of both sides of the ODE.
2. Substitute the initial conditions and use the derivative property to eliminate all derivatives.
3. Solve the resulting algebraic equation for .
4. Apply the inverse Laplace transform to find .

Worked Example 1: First-Order ODE

Solve the initial value problem:

Step 1. Take the Laplace transform of both sides:

Step 2. Substitute :

Step 3. Solve for :

The factor cancels neatly. So .

Step 4. Inverse transform:

We can verify: if for all , then and . The initial condition is also satisfied.

Worked Example 2: Second-Order ODE

Solve the initial value problem:

Step 1. Take the Laplace transform:

Step 2. Substitute initial conditions:

Step 3. Solve for :

Step 4. Recognise the form: since , we have and:

Verification: , . Then . Initial conditions: and . Everything checks out.

Worked Example 3: ODE with a Forcing Function

Solve:

Step 1. Transform both sides:

Step 2. Solve for :

Use partial fractions. Write:

Multiplying through and comparing coefficients gives . So:

Step 3. Inverse transform:

This solution describes the superposition of the natural frequency () and the forcing frequency (), a phenomenon central to understanding resonance and beats.

Why Laplace Transforms Matter

The real power of the Laplace transform lies in how it handles initial conditions automatically. With classical methods (undetermined coefficients, variation of parameters), you first find the general solution and then apply initial conditions. With the Laplace method, the initial conditions are baked into the algebra from the very first step, so the result is the particular solution directly.

Laplace transforms also handle piecewise-defined and discontinuous forcing functions (via the Heaviside step function and the second shifting theorem) and impulsive forces (via the Dirac delta function), situations where classical methods become cumbersome.

Tips for Success

• Memorise the core table. The eight entries above cover the vast majority of problems.
• Practise partial fractions. Inverting a Laplace transform almost always requires decomposing a rational function into simpler pieces.
• Always verify. Substitute your solution back into the original ODE and check that it satisfies both the equation and the initial conditions.
• Watch the signs. The derivative property has a minus sign that is easy to drop.

Frequently Asked Questions

What is the Laplace transform used for?

The Laplace transform converts differential equations into algebraic equations, making them easier to solve. It is widely used in electrical engineering (circuit analysis), control theory (transfer functions), mechanical engineering (vibrations), and physics (heat conduction, wave propagation).

What is the inverse Laplace transform?

The inverse Laplace transform recovers the original time-domain function from its frequency-domain representation . In practice, you decompose into partial fractions and read each piece from a standard table.

Do I need to evaluate the integral to find a Laplace transform?

Rarely. For standard functions, you use the table of known pairs and apply properties like linearity and shifting. You only evaluate the defining integral directly when deriving a new transform pair or proving a result from first principles.

What is the difference between Laplace and Fourier transforms?

Both are integral transforms, but they serve different purposes. The Fourier transform analyses the frequency content of a signal and works with functions defined on all of . The Laplace transform is designed for causal signals (defined for ) and is particularly suited to solving initial value problems. Technically, the Fourier transform is a special case of the Laplace transform evaluated on the imaginary axis .

Can Laplace transforms solve nonlinear differential equations?

Generally, no. The Laplace transform relies on linearity. It works on linear ODEs (constant or variable coefficients, though constant coefficients are far simpler). For nonlinear equations, other techniques such as numerical methods or perturbation theory are needed.