Laplace Transform Calculator

Math Calculus • Differential Equations

Written by STEM Calculators Team Published December 31, 2025 Updated February 24, 2026

5. Laplace Transform Calculator

Compute forward Laplace \(\mathcal{L}\{f(t)\}=F(s)\), inverse Laplace \(\mathcal{L}^{-1}\{F(s)\}=f(t)\), and see detailed steps with your exact inputs substituted.

Mode

Compute:

Syntax: t, s, pi, e, sqrt(), sin(), cos(), exp(), log(). Heaviside: u(t-a) or heaviside(t-a). Dirac delta: delta(t-a) or dirac(t-a).

Input

\(f(t)\): \(F(s)\): Parameters (optional): Transform table:

If you use symbols like a, b in your function, add numeric values above (e.g. a=1). Symbolic transforms still work without values, but numeric checks/plots require them.

DE via Laplace (optional)

For “DE via Laplace”, enter \(y'' + p y' + q y = g(t)\) with initial conditions.

\(p\): \(q\): \(g(t)\): \(y(0)\): \(y'(0)\):

Verification + Graph settings (optional)

Check at \(s^{*}\): Mark on graph \(t^{*}\): Plot: x min: x max: y min: y max: Samples: Engine preference:

Drag to pan; mouse wheel to zoom; double-click resets. Outputs scroll horizontally on small screens.

Ready

Enter an input and click “Solve”.

Graph

Pan/zoom graph — axes/ticks shown

The marked point label shows the actual computed coordinate \((x^{*}, y(x^{*}))\). If parameters are missing, the tool will assume value 1 for plotting only.

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Theory — Laplace Transforms

Definition

The Laplace transform of a function \(f(t)\) (typically \(t\ge 0\)) is \[ \mathcal{L}\{f(t)\}=F(s)=\int_{0}^{\infty} e^{-st}\,f(t)\,dt, \] for values of \(s\) where the integral converges.

The inverse Laplace transform takes \(F(s)\) back to \(f(t)\): \[ \mathcal{L}^{-1}\{F(s)\}=f(t). \]

Core transform table

\(f(t)\)	\(F(s)=\mathcal{L}\{f(t)\}\)
\(1\)	\(\dfrac{1}{s}\)
\(t\)	\(\dfrac{1}{s^2}\)
\(t^n\) (integer \(n\ge 0\))	\(\dfrac{n!}{s^{n+1}}\)
\(e^{at}\)	\(\dfrac{1}{s-a}\)
\(\sin(bt)\)	\(\dfrac{b}{s^2+b^2}\)
\(\cos(bt)\)	\(\dfrac{s}{s^2+b^2}\)
\(u(t-a)\)	\(\dfrac{e^{-as}}{s}\)
\(\delta(t-a)\)	\(e^{-as}\)

\(u(t-a)\) is the Heaviside step (0 for \(t<a\), 1 for \(t\ge a\)). \(\delta(t-a)\) is the Dirac delta (a distribution).

Key properties

Linearity: \[ \mathcal{L}\{af(t)+bg(t)\}=aF(s)+bG(s). \]
Frequency shift: if \(\mathcal{L}\{f(t)\}=F(s)\), then \[ \mathcal{L}\{e^{at}f(t)\}=F(s-a). \]
Time shift: if \(f(t)=u(t-a)\,h(t-a)\), then \[ \mathcal{L}\{f(t)\}=e^{-as}H(s)\quad\text{where}\quad H(s)=\mathcal{L}\{h(t)\}. \]
Derivatives: \[ \mathcal{L}\{y'\}=sY(s)-y(0),\qquad \mathcal{L}\{y''\}=s^2Y(s)-s y(0)-y'(0). \] These are the workhorses for solving IVPs with Laplace transforms.

Inverse Laplace and partial fractions

Many inverse transforms are obtained by rewriting \(F(s)\) into sums of known table entries. A common technique is partial fraction decomposition for rational functions: \[ F(s)=\frac{P(s)}{(s-r_1)(s-r_2)}=\frac{A}{s-r_1}+\frac{B}{s-r_2}. \] Then \[ \mathcal{L}^{-1}\left\{\frac{1}{s-r}\right\}=e^{rt}. \]

For irreducible quadratics, completing the square yields cosine/sine forms: \[ \frac{s-\alpha}{(s-\alpha)^2+\beta^2}\;\leftrightarrow\; e^{\alpha t}\cos(\beta t),\qquad \frac{\beta}{(s-\alpha)^2+\beta^2}\;\leftrightarrow\; e^{\alpha t}\sin(\beta t). \]

Solving DEs via Laplace

For a linear IVP like \[ y'' + p y' + q y = g(t),\quad y(0)=y_0,\;y'(0)=v_0, \] take Laplace transforms: \[ (s^2Y-sy_0-v_0) + p(sY-y_0) + qY = G(s). \] Solve for \(Y(s)\): \[ Y(s)=\frac{G(s)+s y_0+v_0+p y_0}{s^2+p s+q}. \] Then invert to obtain \(y(t)\). The calculator’s “DE via Laplace” mode follows this workflow for common \(g(t)\).

University extension: convolution

If \(F(s)=\mathcal{L}\{f\}\) and \(G(s)=\mathcal{L}\{g\}\), then \[ \mathcal{L}^{-1}\{F(s)G(s)\}=(f*g)(t)=\int_0^t f(\tau)\,g(t-\tau)\,d\tau. \] This is useful when multiplication in the \(s\)-domain is easier than direct inversion.

Frequently Asked Questions

What is the difference between the forward Laplace transform and the inverse Laplace transform?

The forward transform maps a time-domain function f(t) to an s-domain function F(s) using L{f(t)}. The inverse transform maps F(s) back to f(t) using L^-1{F(s)}.

How do I enter a Heaviside step function or a Dirac delta in this Laplace calculator?

Use u(t-a) or heaviside(t-a) for the step function, and delta(t-a) or dirac(t-a) for the impulse. Replace a with your shift value, and define a numerically in Parameters if you want plotting or numeric checks.

Can this calculator solve differential equations using Laplace transforms?

Yes, the DE via Laplace (basic) mode supports IVPs of the form y'' + p y' + q y = g(t) with y(0) and y'(0). It applies Laplace rules for derivatives to solve for Y(s) and then inverts to obtain y(t).

Why does the plot assume missing parameter values are 1?

When a parameter value is not provided, the tool may assume a default value (such as 1) for plotting so a curve can be drawn. For accurate numeric evaluation, define all symbols you want to treat as numbers in the Parameters field.

What does the check at s* or the marked point on the graph mean?

The check value evaluates the computed expression at a specific input (s* for F(s) or t* for f(t)) for verification. The marked point label shows the actual computed coordinate (x*, y(x*)) based on the current plot domain.