What is a Laplace transform?

A Laplace transform converts a time-domain function f(t) into an s-domain function F(s), often making differential equations easier to solve.

Is this Laplace Transform Calculator a full symbolic CAS?

No. It is an educational calculator for common transform pairs, inverse transform patterns, differential equation templates, and circuit applications.

Can Laplace transforms solve differential equations?

Yes. Laplace transforms convert derivatives into algebraic expressions involving s and initial conditions, which can simplify solving initial value problems.

What is the inverse Laplace transform?

The inverse Laplace transform converts an s-domain expression F(s) back into a time-domain function f(t).

How are Laplace transforms used in circuits?

In circuit analysis, Laplace transforms model capacitors and inductors in the s-domain, making RC, RL, and RLC transient responses easier to solve.

All calculators

Laplace Transform Calculator

Calculate common Laplace transforms, inverse transforms, differential equation solutions, and circuit responses with step-by-step explanations.

Background

Laplace transforms convert time-domain problems into s-domain algebra. This calculator focuses on common undergraduate patterns, transform pairs, differential equations, and RC/RL/RLC circuit applications.

Analyze a Laplace transform problem

Choose a mode

Use common transforms, inverse transforms, differential equation examples, or circuit applications.

Common transform input

Enter one supported expression or simple combinations joined by + or -.

Function of t

Inverse transform input

Use one of the supported common s-domain forms.

Function of s

Differential equation solver

This is a template solver for common constant-coefficient homogeneous IVPs.

Equation template

c / forcing constant

Used for y' + ay = c.

y(0)

y'(0)

Circuit application

See how Laplace transforms turn differential equations into algebraic transfer functions.

Circuit type

Resistance, R (Ω)

Capacitance, C (F)

Inductance, L (H)

Step input, V₀ (V)

Supported / not supported yet

This calculator is intentionally focused on common course patterns, not arbitrary symbolic algebra.

Supported inputs

Forward: t^2, e^(3t), sin(5t), cos(2t), sinh(4t), simple sums, and scalar multiples.

Supported inverses

Inverse: 1/(s+3), 5/(s^2+25), s/(s^2+4), and simple power forms like 2/s^3.

Not supported yet

No full CAS, arbitrary partial fractions, convolutions, piecewise functions, Dirac delta, Heaviside step functions, or long symbolic expressions yet.

Quick examples (click to see result)

Options

Show step-by-step solution

Show s-domain / pole visual

Show transform pair table

Show “why this works” explanation

Result

Copied!

No result yet. Choose a mode, enter a supported expression, then click Calculate Laplace Transform.

How to use this calculator

Choose common transform, inverse transform, differential equation, or circuit application mode.
Enter a supported expression or select one of the built-in templates.
Click Calculate Laplace Transform to see the result, rule used, steps, and visual.
Use quick examples to test common homework patterns before entering your own expression.

Formula & transform pairs used

Definition: L{f(t)} = ∫₀∞ e^(-st)f(t)dt

Constant: L{1} = 1/s

Power: L{tⁿ} = n!/sⁿ⁺¹

Exponential: L{e^(at)} = 1/(s-a)

Sine: L{sin(at)} = a/(s²+a²)

Cosine: L{cos(at)} = s/(s²+a²)

Example Problems & Step-by-Step Solutions

Example 1: Transform a power

For f(t)=t², use L{tⁿ}=n!/sⁿ⁺¹. With n=2, the result is 2/s³.

Example 2: Inverse sine transform

5/(s²+25) matches a/(s²+a²) with a=5, so the inverse transform is sin(5t).

Example 3: Solve an SHM-style IVP

For y''+4y=0, y(0)=2, y'(0)=0, the Laplace method gives Y(s)=2s/(s²+4) and y(t)=2cos(2t).

Example 4: First-order IVP

For y' + 3y = 6, y(0)=2, the steady value is 6/3=2, so y(t)=2. The Laplace setup is (s+3)Y - 2 = 6/s.

Why Laplace transforms help

A Laplace transform turns differentiation into multiplication by s plus initial-condition terms. That makes many differential equations and circuit problems easier to solve because the hard time-domain equation becomes algebra in the s-domain.

FAQ

Is this a full symbolic CAS?

No. It supports common transform pairs, simple combinations, selected inverse forms, differential equation templates, and circuit templates.

Can it solve differential equations?

Yes, for common undergraduate constant-coefficient initial value problem templates.

What does the s-domain mean?

The s-domain is a transformed domain where time-domain derivatives become algebraic terms involving s.