Laplace Transform Calculator

Math Calculus • Multivariable Calculus

Written by STEM Calculators Team Published June 28, 2026 Updated June 28, 2026

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Laplace Transforms — Theory

1. Definition

The Laplace transform changes a time-domain function \(f(t)\) into an \(s\)-domain function \(F(s)\).

\[ \mathcal{L}\{f(t)\}=F(s)=\int_{0}^{\infty}e^{-st}f(t)\,dt. \]

The inverse Laplace transform changes \(F(s)\) back into \(f(t)\):

\[ \mathcal{L}^{-1}\{F(s)\}=f(t). \]

2. Why Laplace transforms are useful

Laplace transforms turn many differential equations into algebraic equations. Derivatives of \(y(t)\) become expressions involving \(Y(s)\), \(y(0)\), and \(y'(0)\).

\[ \mathcal{L}\{y'\}=sY(s)-y(0). \]

\[ \mathcal{L}\{y''\}=s^2Y(s)-s\,y(0)-y'(0). \]

This is why Laplace transforms are especially useful for initial value problems.

3. Common transform table

Time-domain function	Laplace transform	Comment
\(\displaystyle 1\)	\(\displaystyle \frac{1}{s}\)	Constant function.
\(\displaystyle t\)	\(\displaystyle \frac{1}{s^2}\)	First power of \(t\).
\(\displaystyle t^n\)	\(\displaystyle \frac{n!}{s^{n+1}}\)	For integer \(n\ge 0\).
\(\displaystyle e^{at}\)	\(\displaystyle \frac{1}{s-a}\)	Exponential growth or decay.
\(\displaystyle \sin(bt)\)	\(\displaystyle \frac{b}{s^2+b^2}\)	Sine transform.
\(\displaystyle \cos(bt)\)	\(\displaystyle \frac{s}{s^2+b^2}\)	Cosine transform.
\(\displaystyle u(t-a)\)	\(\displaystyle \frac{e^{-as}}{s}\)	Unit step beginning at \(t=a\).
\(\displaystyle \delta(t-a)\)	\(\displaystyle e^{-as}\)	Impulse at \(t=a\).

4. Linearity

The Laplace transform is linear. This means constants can be pulled out, and sums can be transformed term by term.

\[ \mathcal{L}\{af(t)+bg(t)\}=aF(s)+bG(s). \]

Example:

\[ \mathcal{L}\{3\sin(2t)+5\cos(2t)\} = 3\frac{2}{s^2+4}+5\frac{s}{s^2+4}. \]

5. Frequency shift

Multiplying by \(e^{at}\) shifts the \(s\)-variable.

\[ \mathcal{L}\{e^{at}f(t)\}=F(s-a). \]

In particular:

\[ \mathcal{L}\{e^{at}\}=\frac{1}{s-a}. \]

6. Time shift and the Heaviside function

The Heaviside step function \(u(t-a)\) turns on at \(t=a\). It is often used to model switching.

\[ u(t-a)= \begin{cases} 0,&t

A basic table entry is:

\[ \mathcal{L}\{u(t-a)\}=\frac{e^{-as}}{s}. \]

More generally, if \(h(t-a)\) starts at \(t=a\), then:

\[ \mathcal{L}\{u(t-a)h(t-a)\}=e^{-as}H(s). \]

7. Dirac delta impulse

The Dirac delta \(\delta(t-a)\) represents an ideal impulse at \(t=a\). Its transform is:

\[ \mathcal{L}\{\delta(t-a)\}=e^{-as}. \]

This is useful for impulse forcing in differential equations.

8. Inverse Laplace transforms

To find an inverse Laplace transform, rewrite \(F(s)\) so that it matches table entries.

\[ \mathcal{L}^{-1}\left\{\frac{1}{s-a}\right\}=e^{at}. \]

\[ \mathcal{L}^{-1}\left\{\frac{s}{s^2+b^2}\right\}=\cos(bt). \]

\[ \mathcal{L}^{-1}\left\{\frac{b}{s^2+b^2}\right\}=\sin(bt). \]

9. Partial fractions

Rational functions often need partial fractions before using the inverse table. For example:

\[ \frac{1}{(s+1)(s+2)} = \frac{1}{s+1}-\frac{1}{s+2}. \]

Therefore:

\[ \mathcal{L}^{-1}\left\{ \frac{1}{(s+1)(s+2)} \right\} = e^{-t}-e^{-2t}. \]

10. Solving differential equations with Laplace transforms

Consider:

\[ y''+p y'+q y=g(t), \qquad y(0)=y_0,\quad y'(0)=v_0. \]

Take Laplace transforms of both sides:

\[ \left(s^2Y-sy_0-v_0\right) +p\left(sY-y_0\right) +qY = G(s). \]

Collect the \(Y(s)\) terms:

\[ \left(s^2+ps+q\right)Y(s) = G(s)+sy_0+v_0+py_0. \]

Then solve for \(Y(s)\):

\[ Y(s)= \frac{ G(s)+sy_0+v_0+py_0 }{ s^2+ps+q }. \]

11. Worked example: \(y''+y=0\)

Solve:

\[ y''+y=0, \qquad y(0)=1, \qquad y'(0)=0. \]

Take Laplace transforms:

\[ s^2Y-s+Y=0. \]

Factor \(Y\):

\[ (s^2+1)Y=s. \]

Solve for \(Y(s)\):

\[ Y(s)=\frac{s}{s^2+1}. \]

Use the inverse transform:

\[ \mathcal{L}^{-1}\left\{\frac{s}{s^2+1}\right\}=\cos(t). \]

Therefore:

\[ y(t)=\cos(t). \]

12. Formula summary

Concept	Formula	Meaning
Laplace transform	\(\displaystyle F(s)=\int_0^\infty e^{-st}f(t)\,dt\)	Converts time-domain functions to \(s\)-domain functions.
Inverse transform	\(\displaystyle \mathcal{L}^{-1}\{F(s)\}=f(t)\)	Returns from \(s\)-domain to time-domain.
Linearity	\(\displaystyle \mathcal{L}\{af+bg\}=aF+bG\)	Transforms sums term by term.
First derivative	\(\displaystyle \mathcal{L}\{y'\}=sY-y(0)\)	Converts one derivative into algebraic form.
Second derivative	\(\displaystyle \mathcal{L}\{y''\}=s^2Y-sy(0)-y'(0)\)	Main rule for second-order IVPs.
DE solution form	\(\displaystyle Y(s)=\frac{G(s)+sy_0+v_0+py_0}{s^2+ps+q}\)	For \(y''+py'+qy=g(t)\).

13. Common mistakes

Forgetting initial values: derivative transforms include \(y(0)\) and \(y'(0)\).
Using \(t\) in the transform result: \(F(s)\) should be written in the \(s\)-domain.
Using \(s\) in the inverse result: \(f(t)\) should be written in the time domain.
Forgetting linearity: transform sums one term at a time.
Missing partial fractions: many inverse transforms require rewriting rational functions first.
Confusing sine and cosine entries: \(\sin(bt)\) gives \(b/(s^2+b^2)\), while \(\cos(bt)\) gives \(s/(s^2+b^2)\).
Ignoring graph units: axis tick labels should show numerical values together with units.

Frequently Asked Questions

What is the Laplace transform?

The Laplace transform changes a time-domain function f(t) into an s-domain function F(s) using the integral from 0 to infinity of e^{-st}f(t) dt.

What is the inverse Laplace transform?

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

What table entries does this calculator support?

It supports common entries such as constants, t^n, exponentials, sine, cosine, unit step functions, impulses, and selected rational inverse forms.

Does it support partial fractions?

Yes. It supports selected simple rational forms such as 1/((s+1)(s+2)) and similar products of two distinct linear factors.

How does the DE mode work?

The DE mode applies the Laplace transform to y'', y', and y, solves algebraically for Y(s), and then graphs the time-domain response.

What does the s-domain graph show?

It shows values of F(s) over the displayed x-window when F(s) is finite there.

Do the graph axes show units?

Yes. Axis tick labels include both numerical values and the selected units, such as 2 time or 1 amplitude.

Laplace Transform Calculator

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